![Chapter 1
COUPLED NONLINEAR OSCILLATORS
1.1 The Role of Synchronization
Synchronization occurs when oscillatory (or repetitive) systems via some kind
of interaction adjust their behaviors relative to one another so as to attain a
state where they work in unison. An essential aspect of many of the games we
play as children is to teach us to coordinate our motions. We skip and learn to
jump in synchrony with the swinging rope. We run along the beach and learn
to avoid the waves that role ashore, and we take dancing lessons to learn to
move in step with the music.
One of the main problems in the swimming class is to learn to breath in syn-
chrony with the strokes. Not necessarily one-to-one, as there are circumstances
where it is advantageous to take two (or more) strokes per inhalation. However,
the phase relations must be correct if not to drown. In much the same way, a
horse has different forms of motion (such as walk, trot, and gallup), and each of
these gaits corresponds to a particular rhythm in the movement of its legs [1,2].
At the trotting course, the jockey tries to keep the horse in trot to the highest
possible speed. In its free motion, however, a horse is likely to choose the mode
that is most comfortable to it (and, perhaps, least energy demanding). As the
speed increases the horse will make transitions from walk to trot and from trot
to gallup.
Synchronization is a universal phenomenon in nonlinear systems [3]. Well-
known examples are the synchronization of two (pendulum) clocks hanging on a
l](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-16-320.jpg)
![2 Chaotic Synchronization: Applications to Living Systems
wall, and the synchronization of the moon's rotation with its orbital motion so
that the moon always turns the same side towards the earth. A radio receiver
functions by synchronizing its internal oscillator with the period of the radio
wave so that the difference, i.e. the transmitted signal, can be detected and
converted into sound. A microwave emitting diode is placed in a cavity of a spe-
cific form and size to make it synchronize with a particular resonance frequency
of the cavity. In a previous book [4] we presented results on synchronization
of coupled thermostatically controlled radiators and coupled household refrig-
erators. Synchronization can also be observed between coupled laser systems
and coupled biochemical reactors, and it is clear that one can find thousands
of other examples in engineering and physics. At the assembly line one has to
ensure an effective synchronization of the various processes for the production
to proceed in an efficient manner, and engineers and scientists over and over
again exploit the technique of modulating (or chopping) a test signal in order
to benefit from the increased sensitivity of phase detection.
The history of synchronization dates back at least to Huygens' observations
some 300 years ago [5], and both the history and the basic theory are recapitu-
lated in a significant number of books and articles [6, 7]. For regular (e.g., limit
cycle) oscillators, synchronization implies that the periodicities of the interact-
ing systems precisely coincide and that differences in phase remain constant. In
the presence of noise (or for chaotic systems) one can weaken the requirements
such that the periodicities only have to coincide on average, and the phase
differences are allowed to move within certain bounds. One may also accept
occasional phase slips, provided that they do not occur too often [8].
One-to-one synchronization is only a simple manifestation of a much more
general phenomenon, also known as entrainment, mode locking, or frequency
locking. In nonlinear systems, a periodic motion is usually accompanied by a
series of harmonics at frequencies of p times the fundamental frequency, where
p is an integer. When two nonlinear oscillators interact, mode locking may
occur whenever a harmonic frequency of one mode is close to a harmonic of
the other. As a result, nonlinear oscillators tend to lock to one another so
that one subsystem completes precisely p cycles each time the other subsystem
completes q cycles, with p and q as integers [9, 10]. An early experience with
this type of phenomenon is the way one excites a swing by forcing it at twice its
characteristic frequency, i.e., we move the body through two cycles of a bending
and stretching mode for each cycle of the swing. A similar phenomenon is](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-17-320.jpg)
![Coupled Nonlinear Oscillators 3
utilized (in optics, electronics, etc.) in a wide range of so-called parametric
devices.
Contrary to the conventional assumption of homeostasis, many physiological
systems are unstable and operate in a pulsatile or oscillatory mode [10, 11]. This
s the case, for instance, for the production of luteinizing hormone and insulin
hat are typically released in two-hour intervals [4]. In several cases it has been
>bserved that the cellular response to a pulsatile hormonal signal is stronger
han the response to a constant signal of the same magnitude, suggesting that
he oscillatory dynamics plays a role in the control of the system [12]. Hormonal
elease processes may also become synchronized, and it has been reported,
or instance, that the so-called hot flashes that complicate the lives of many
vomen during menopause are related to the synchronized release of 5-7 different
lormones [13, 14].
The beating of the heart, the respiratory cycle, the circadian rhythm, and the
avarian cycle are all examples of more or less regular self-sustained oscillations.
The ventilatory signal is clearly visible in spectral analyses of the beat-to-beat
variability of the heart signal, and in particular circumstances the two oscillators
may lock together so that, for instance, the heart beats three or four times for
each respiratory cycle [15]. The jet lag that we experience after a flight to a
different time zone is related to the synchronization of our internal (circadian)
rhythm to the local day-and-night cycles, and it is often said that women can
synchronize their menstrual cycles via specific scents (pheromones) if they live
close together.
Rhythmic and pulsatile signals are also encountered in intercellular commu-
nication [16]. Besides neurons and muscle cells that communicate by trains
of electric pulses, examples include the generation of cyclic AMP pulses in
slime mold cultures of Dictostelium discoideum [17] and the newly discovered
synchronization of the metabolic processes in suspensions of yeast cells [18].
Synchronization of the activity of the muscle cells in the heart is necessary for
the cells to act in unison and produce a regular contraction. Similarly, groups
of nerve cells must synchronize to produce the characteristic rhythms of the
brain or to act as pacemakers for the glands of the hormonal systems [19]. On
the other hand, it is well-known that synchronization of the electrical activity
of larger groups of cells in the brain plays as essential role in the development
of epileptic seizures [20].
However, nonlinear oscillators may also display more complicated forms of](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-18-320.jpg)
![4 Chaotic Synchronization: Applications to Living Systems
dynamics, and an interesting question that arises over and over again in the bio-
logical sciences concerns the collective behavior of a group of cells or functional
units that each display strongly nonlinear phenomena [21].
The human kidney, for instance, contains of the order of one million func-
tional units, the nephrons. In order to protect its function, the individual
nephron disposes of a negative feedback regulation by which it can control the
incoming blood flow. However, because of the delay associated with the flow
of fluid through the nephron, this regulation tends to be unstable and pro-
duce self-sustained oscillations in the various pressures and concentrations [22].
If the arterial blood pressure is high enough, the pressure oscillations in the
nephron may become irregular and chaotic [23]. Neighboring nephrons interact
with one another through signals that propagate along the afferent arterioles
(incoming blood vessels) and, as experiments show, this interaction can lead
to a synchronization of the regular pressure oscillations for adjacent nephrons
[24].
Fig 1.1. Pressure variations in two
neighboring nephrons for a hyperten-
sive rat. Note that there is a certain
degree of synchronization between the
irregular (chaotic) signals. This syn-
chronization is found to arise from in-
teractions between the nephrons and
not from common external influences.
'"*" 0 500 1000
Time (sec)
It is obviously of interest to examine to what extent similar synchronization
phenomena are manifest in the irregular oscillations at higher blood pressures.
Figure 1.1 shows an example of the chaotic pressure variations that one can
observe (in the proximal tubule) for neighboring nephrons in a hypertensive
rat. Although the two signals are strongly irregular, one is tempted to admit
that there is a certain degree of synchronization: The most pronounced maxima
and minima in the pressure variations occur almost simultaneously. Figure 1.2](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-19-320.jpg)
![Coupled Nonlinear Oscillators 5
shows a scanning electron microscope picture of the interaction structure for
a couple of nephrons. Here, one can see how the common interlobular artery
(IA) branches into separate afferent arterioles (af) for the two nephrons. The
two ball-formed bundles are the capillary systems (the glomeruli) of the two
nephrons. Here, blood constituents like water and salts are filtered into the
tubular system of the nephrons and the remaining blood passes out through the
efferent arterioles (e/). See Fig. 9.1 for a more detailed sketch of the structure
of the nephron.
We would like to understand how the interaction between the nephrons in-
fluence the overall functioning of the kidney. Will there be circumstances,
for instance, where the coupling produces a global synchronization of all the
nephrons or will we see the formation of clusters of nephrons in different syn-
chronization states? Will transitions between different states of synchronization
play a role in the regulation of the kidney or will such transitions be related to
the development of particular diseases?
Similarly, each of the insulin producing /3-cells of the pancreas exhibits a
complicated pattern of oscillations and bursts in its membrane potential [25].
Presumably through their relation to the exchange of calcium between the cell
and its surroundings, these bursts control the release of insulin. The /3-cells
are arranged in a spiral structure along capillaries and small veins. Via insulin
receptors in the cell membrane, each cell can thus react to the release of insulin
from cells that are upstream to it. At the same time, the /3-cells are coupled
Fig 1.2. Scanning electron microscope pic-
ture of the arteriolar system for a couple of
adjacent nephrons. The nephrons are as-
sumed to interact with one another via mus-
cular contractions that propagate along the
afferent arterioles (af).](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-20-320.jpg)
![6 Chaotic Synchronization: Applications to Living Systems
via gap junctions through which ions and small molecules can pass from cell to
cell. Again it is of interest to understand how the collective behavior of a group
of cells is related to the dynamics of individual cells. Experiments indicate that
there will be waves of synchronization moving across larger groups of cells in
an islet of Langerhans [26].
In the economic realm, each individual production sector with its charac-
teristic capital life time and inventory coverage parameters tends to exhibit
an oscillatory response to changes in the external conditions [27]. Overreac-
tion, time delays, and reinforcing positive feedback mechanisms may cause the
behavior to become destabilized and lead to complicated nonlinear dynamic
phenomena. The sectors interact via the exchange of goods and services and
via the competition for labor and other resources. A basic problem for the
establishment of a dynamic macroeconomic theory is therefore to describe how
the various interactions lead to a more or less complete entrainment of the
sectors [28].
The problems associated with chaotic synchronization have also attracted a
considerable interest in the fields of electronics and radio engineering. Here, the
attention centers around the possibilities of developing new types of commu-
nication systems that exploit the particular properties of deterministic chaos
[29, 30, 31]. Important questions that arise in this connection pertain to the
sensitivity of the synchronized state to noise or to a parameter mismatch be-
tween the interacting oscillators. Other questions relate to the behavior of the
coupled system, once the synchronization breaks down, and to the initial con-
ditions for which entrainment can be attained. It is a problem of considerable
interest, whether or not one can mask a message by mixing it with a chaotic
signal [32].
In order to discuss some of the problems that arise in connection with chaotic
synchronization we shall apply a variety of different simple mathematical mod-
els. We start in Chapters 2 and 3 by considering a system of two (or three)
coupled logistic maps. This leads us to a discussion of the conditions for syn-
chronization in systems of coupled Rossler oscillators (Chapter 4) and in a
system of two (nearly) identical /3-cells (Chapter 5). In this connection we
show that a /3-cell has regions of chaotic dynamics between the different states
of periodic bursting. Towards the end of the book the analysis will lead us to
consider clustering in systems of many coupled chaotic oscillators (Chapter 8)
and to examine interacting coherence resonance (i.e., stochastically excited) os-](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-21-320.jpg)
![Coupled Nonlinear Oscillators 7
dilators (Chapter 10). On the way we shall discuss the characteristics of chaotic
phase synchronization (Chapter 6) and use the obtained results to examine ex-
perimental data for the tubular pressure variations in neighboring nephrons
(Chapter 9). Let us start, however, by discussing some of the characteristic
signatures of synchronization in regular and chaotic systems. Thereafter, we
shall use a model of two interacting capital producing sectors of the economy
[28] to recall some of the basic concepts of the classical synchronization theory
[33] and to illustrate the role of synchronization in macroeconomic systems.
1.2 Synchronization Measures
Let us consider some of the phenomena that one can observe in connection with
chaotic phase synchronization [34, 35]. This is the type of synchronization that
we expect to find between two coupled chaotic oscillators with different param-
eters such as, for instance, between neighboring nephrons in a hypertensive rat.
The idea is to focus on the similarities between chaotic phase synchronization
and the synchronization phenomena we know for regular oscillators. Among of
the questions we would like to discuss are: What are the signatures of chaotic
phase synchronization? Can we use similar diagnostic tools as we use for regular
oscillators? What are the main bifurcation scenarios? First, however, we should
perhaps recall some of the characteristics of the synchronization mechanism for
regular oscillators [4, 9, 10].
From a mathematical point of view we understand the synchronization of
two periodic oscillators as a transition from quasiperiodic motion to regular
periodic behavior for the system as a whole. The quasiperiodic behavior is
usually described as the motion on a torus. This motion is characterized by the
presence of two incommensurate periods, asscociated with the motions of the
individual oscillators. As coupling between the oscillators is introduced, both
oscillators adjust their motions in response to the motion of the other, and
when the coupling becomes strong enough a transition typically occurs where
the two periods start to coincide. In the absence of coupling, the phase of each
oscillator is a neutrally stable variable. There are no mechanisms that act to
correct for a shift in phase. The amplitude, on the other hand, is controlled by
a balance between instability and nonlinearity, and dissipation leads to a rapid
decay of any pertubation of the amplitude. Hence, we conclude that mutual
phase adjustments will be more significant than amplitude modulation [3]](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-22-320.jpg)
![8 Chaotic Synchronization: Applications to Living Systems
At least for relatively small coupling strengths, the synchronization takes
place via a saddle-node bifurcation [33]. On the surface of the torus a stable
(node) and an unstable (saddle) cycle simultaneously emerge. Under variation
of a control parameter (for instance, a parameter that controls the uncoupled
period of one of the oscillators), the two cycles move away from one another
along the torus surface to meet again and become annihilated on the opposite
side. As a result, the synchronized state exists in a finite range of the control
parameter.
The typical situations where synchronization occurs are mutually coupled
oscillators and periodically forced oscillators. Glass and Mackey [10] have dis-
cussed, for instance, how different forms of synchronization can be observed for
periodically forced chicken heart cells. Sturis et al. [36] have described how the
release of insulin from the pancreas in normal subjects can be synchronized to
an external variation in the supply of glucose, and Bindschadler and Sneyd [37]
have described how oscillations in the intracellular concentrations of calcium
in biological cells can be synchronized via the exchange of ions through the
gap junctions. Under such conditions, the relevant control parameters are the
frequency mismatch Aw for the uncoupled oscillators and the coupling strength
(or forcing amplitude) K.
The phenomenon of synchronization can be described from different perspec-
tives. If we look at the Fourier spectra of the oscillations, synchronization can
be seen as a characteristic evolution of the amplitudes and frequencies of the
spectral components. On the other hand, in terms of a phase space analysis,
the synchronization mechanisms are the possible ways of transition from an
ergodic (or nonresonant) two-dimensional torus (which, as mentioned above, is
the image of quasiperiodic behavior) to a limit cycle, being the image of periodic
oscillations.
Figure 1.3 shows the typical structure of the 1:1 synchronization regime
[38, 39]. To be specific, let us talk about the case of forced synchronization.
The representative ways on the (Aw, .^-parameter plane are denoted with
arrows:
Route A: At weak coupling and a relatively small parameter mismatch the
onset of resonance on the two-torus corresponds to the crossing of one of the
saddle-node bifurcation lines SN. At this point, a pair of limit cycles (a sta-
ble and a saddle cycle) emerge on the torus surface. In terms of the Fourier
spectra, this transition can be diagnosed from the approach and final merging](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-23-320.jpg)
![10 Chaotic Synchronization: Applications to Living Systems
Here, one observes a region of bistability, where the stable synchronous solu-
tion and the stable nonresonant torus coexist. Under variation of the control
parameters, this bistability manifests itself in terms of a hysteretic behavior.
Most of the published work on chaotic phase synchronization refers to the
case where the chaotic dynamics has appeared through a cascade of period-
doubling bifurcations. This type of chaotic dynamics has a characteristic struc-
ture which manifests itself both in the rotation of the trajectory around some
center point (Fig. 1.4(a)) and in the presence of a characteristic time scale which
can be easily measured from the power spectrum (see Fig. 1.4(b)). These fea-
tures of period-doubling chaos are important in relation to the problem of phase
synchronization because they make it possible to introduce a simple measure
of the instantaneous phase of the chaotic oscillations and to consider the mean
return time to some Poincare secant as representative of the internal rhythm of
the dynamics.
(b)
0 0.0 0.5 1.0 1.5 2.0
(0
Fig 1.4. The chaotic attractor of the Rossler model demonstrates the typical features of period
doubling chaos. The phase trajectory rotates around some center in the phase space projection
(a), and one can see the corresponding peak of the fundamental frequency in the power spectrum
(b).
A relatively small number of model dynamical systems have served an im-
portant role in the investigation of period-doubling chaos. The same is true
for the problem of chaotic phase synchronization. Here, we can mention the
Rossler model [40, 41], the Chua circuit [42, 43], and the Anishchenko-Astakhov
generator [35, 44]. All of these models are three-dimensional oscillators demon-
strating the period-doubling route to chaos, with the high-periodic and chaotic](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-25-320.jpg)
![Coupled Nonlinear Oscillators 11
regimes located around a single unstable equilibrium point of saddle-focus type.
During the period-doubling cascade, no additional complexity occurs, and these
models thus describe a generic case.
By way of example let us base our discussion on the Rossler model [45]:
x = —u>y — z,
y = UJX + ay,
z = (3 + z(x-n), (1.1)
where a, (3, and x are control parameters, and UJ defines the characteristic
frequency of the oscillations. The chaotic dynamics of this model is well-studied.
There are two possible types of chaotic attractor. With i increasing, a cascade
of period-doubling bifurcations leads to the emergence of chaos. This is called
"spiral chaos". This type of chaos is illustrated in Fig. 1.4(a). If /x increases
further, the more complicated chaotic motion referred to as "screw chaos" can
be observed [44].
To study the synchronization mechanisms we rewrite the model (1.1) as a
nonautonomous system:
x = —ujy — z + Ksinujft,
y = UJX + ay,
z = p + z{x-fi), (1.2)
where K is the amplitude of a external harmonic drive, and the forcing fre-
quency ojj is fixed at 1.0.
As illustrated in Fig. 1.4 (b) there is usually a well developed peak in the
power spectrum of period-doubling chaos at a frequency close to the frequency
of the period-one limit cycle from which the chaotic dynamics has originated.
Following the changes in the peak position, one can find an interval for the con-
trol parameter UJ where the peak frequency UJQ is in a rational relation with the
frequency of the forcing signal ujf. This method works well both for numerical
simulations and for full scale experiments. The first observation of frequency
locking for chaotic oscillations was made with a similar approach [34, 35].
Figure 1.5 illustrates the locking of the fundamental frequency of the chaotic
oscillations for the nonautonomous Rossler model. For UJ G [0.922; 0.929], a
1:1 locking region is observed. Chaotic oscillations inside and outside this re-](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-26-320.jpg)
![12 Chaotic Synchronization: Applications to Living Systems
Fig 1.5. Frequency locking region for the
chaotic oscillations in the nonautonomous
Rossler model with a = 0.2, /? = 0.2, // =
4.0, and K = 0.02.
""'6.920 0.925 0.930
CO
gion can be classified as synchronous and asynchronous, respectively. This is
confirmed by Fig. 1.6 where the Poincare sections for both cases are plotted.
It is interesting to note how Fig. 1.5 reproduces the variation that one ob-
serves in connection with the synchroniztion of a regular oscillator, forced by a
periodic signal. In the interval of chaotic phase synchronization (0.922 < u> <
0.929), the characteristic frequency of the chaotic oscillations coincides precisely
with the forcing frequency. This corresponds to the 1:1 step of the well-know
devil's staircase for regular oscillations [4, 9, 10]. On both sides of the synchro-
nization interval, the characteristic frequency of the chaotic oscillations shifts
away from the forcing frequency. This region of asynchronous chaos replaces
the quasiperiodic region for regular oscillators.
Inspection of Fig. 1.6(b) shows how the structure of the Poincare section for
the asynchronous chaotic dynamics reproduces the characteristic structure of
the phase space projection for the autonomous Rossler oscillator (see Fig. 1.4).
This implies that our stroboscopic measurements (the Poincare section for a
forced system) catch the Rossler oscillator more or less at random in all different
positions along its trajectory. In the synchronized chaotic case (Fig. 1.6(a)), on
the other hand, although there is a certain scatter in the position of the Rossler
system in the stroboscopic map, the system is always found in a relatively
narrow region of phase space. Whenever the forcing signal has completed a full
period, the synchronized Rossler oscillator is back to nearly the same position
in phase space.
In the following chapters (particularly in Chapter 6) we shall return to the
problem of chaotic phase synchronization to discuss additional signatures of
synchronization (e.g., the variation of the Lyapunov exponents) and to examine](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-27-320.jpg)
![Coupled Nonlinear Oscillators 13
10.0 | (a)
5.0
-5.0 •
-10.0
-10.0 -5.0 0.0 5.0 10.0 -10.0 -5.0 0.0 5.0 10.0
X X
Fig 1.6. Phase projections of the Poincare section in the nonautonomous Rossler model for
synchronous (a) and asynchronous (b) chaos at fj, = 4.0, a = 0.2, /? = 0.2, and K = 0.02.
the bifurcations involved in the transition to chaotic phase synchronization. We
shall also show that chaotic phase sychronization can develop along the same
routes as we have illustrated for regular oscillations in Fig. 1.3. However, let us
complete the present chapter with a discussion of the role of synchronization
in economic systems. This is an area of research which appears so far to have
attracted far too little attention.
1.3 Mode-Locking of Endogenous Economic Cycles
Macroeconomic models normally aggregate the individual firms of the economy
into sectors with similar products, manufacturing processes and decision rules.
Sometimes, only a single sector is considered (e.g., [46, 47]). This simplification
is justified on pragmatic grounds by noting that it is impractical to portray sep-
arately all the firms in an industry or all the products on the market, and by
arguing that the phenomena of interest are captured in sufficient detail by the
aggregate formulation [48, 49]. Nevertheless, there are instances where aggre-
gation is not justified. Economic sectors associated with the new information
and communication technologies, for example, may show rapid growth while
other, more traditional sectors show little increase or even decline. Here, dis-
aggregation is required to provide a proper perspective of future developments.
The oscillatory patterns that one can observe in many economic variables also
display widely different periodicities. Commodity cycles in pork prices and
slaughter rates typically exhibit a period of 3-4 years while similar cycles for
-10.0](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-28-320.jpg)
![14 Chaotic Synchronization: Applications to Living Systems
chicken and cattle have periodicities of about 30 months and 15 years, re-
spectively [50]. These cycles are related to the feeding periods for the various
animals. In other sectors, such as the auto industry or the tanker market, one
can find periodicities that relate to the lifetime of the capital [4, 51].
Obviously, the various sectors interact with one another. However, the eco-
nomic modeling literature is weak in providing guidelines for appropriate aggre-
gation of dynamic systems, particularly when there are significant interactions
between the individual entities. Models of capital investment, for instance, typ-
ically represent the average lead time and lifetime of the plant and equipment
used by each firm. In reality there are many types of plant and equipment
acquired from many vendors operating with a wide range of lead times. In
response to changes in its external conditions, each firm will generate cyclic
behaviors whose frequency, damping, and other properties are determined by
the parameters characterizing the particular mix of lead times and lifetimes the
firm faces. Because the individual firms are coupled to one another via the
input-output structure of the economy, each acts as a source of perturbations
on the others.
How do the different lifetimes and lead times of plant and equipment affect
the frequency, phase, amplitude, and coherence of economic cycles? How valid
is aggregation of individual firms into single sectors for the purpose of studying
macroeconomic fluctuations? Here, the issue of coherence or synchronization
becomes important. The economy as a whole experiences aggregate business
fluctuations of various frequencies from the short-term business cycle to the
long-term Kondratieff cycle [4, 27]. Yet why should the oscillations of the
individual firms move in phase so as to produce an aggregate cycle? Given the
distribution of parameters among individual firms, why do we observe only a
few distinct cycles rather than cycles at all frequencies - cycles which might
cancel out at the aggregate level?
A common approach to the question of synchronization in economics is to
assume that fluctuations in aggregates such as gross domestic product or un-
employment arise from external shocks, for example sudden changes in resource
supply conditions or variations in fiscal or monetary policy [52]. Forrester [53]
suggested instead that synchronization could arise from the endogenous inter-
action of multiple nonlinear oscillators, i.e., that the cycles generated by indi-
vidual firms become reinforced and entrained with one another. Forrester also
proposed that such entrainment could account for the uniqueness of the eco-](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-29-320.jpg)
![Coupled Nonlinear Oscillators 15
nomic cycles. Oscillatory tendencies of similar periodicity in different parts of
the economy would be drawn together to form a subset of distinct modes, such
as business cycles, construction cycles, and long waves, and each of these modes
would be separated from the next by a wide enough margin to avoid synchro-
nization. Until recently, however, these suggestions have not been subjected to
rigorous analysis.
Roughly speaking, synchronization occurs because the nonlinear structure of
the interacting parts of a system creates forces that " nudge" the parts of the sys-
tem into phase with one another. As described by Huygens [5], two mechanical
clocks, hanging on the same wall, are sometimes observed to synchronize their
pendulum movements. Each clock has an escapement mechanisms, a highly
nonlinear mechanical devise, that transfers power from the weights to the rod
of the pendulum. When a pendulum is close to the position where the escape-
ment releases, a small disturbance, such as the faint click from the release of the
adjacent clock's escapement, may be enough to trigger the release. Hence, the
weak coupling of the clocks, through vibrations in the wall, can bring individual
oscillations into phase, provided that the two uncoupled frequences are not too
different (see [3] for a more complete discussion of Huygens results).
We have previously described how mode-locking and other nonlinear dy-
namic phenomena arise in a simple model of the economic long wave [4, 27, 54].
As described by Sterman [55], the model explains the long wave as a self-
sustained oscillation arising from instabilities in the ordering and production
of capital. An increase in the demand for capital leads to further increases
through the investment accelerator or "capital self-ordering", because the ag-
gregate capital-producing sector depends on its own output to build up its stock
of productive capital. Once a capital expansion gets under way, self-reinforcing
processes sustain it beyond its long-term equilibrium, until production catches
up with orders. At this point, however, the economy has acquired considerable
excess capital, forcing capital production to remain below the level needed for
replacement until the excess has been fully depreciated, and room for a new
expansion has been created.
The concern of the present discussion is the model's aggregation of capital
into a single type. The real economy consists of many sectors employing dif-
ferent kinds of capital in different amounts. Parameters, such as the average
productive life of capital and the relative amounts of different capital compo-
nents employed, may vary from sector to sector. In isolation, the buildings](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-30-320.jpg)
![16 Chaotic Synchronization: Applications to Living Systems
industry may show a temporal variation significantly different from that of, for
instance, the machinery industry.
An early study by Kampmann [56] took a first step in this direction by disag-
gregating the simple long-wave model into a system of several capital producing
sectors with different characteristics. Kampmann showed that the multi-sector
system could produce a range of different behaviors, at times quite different from
the original one-sector model. The present analysis provides a more formal ap-
proach, using a two-sector model. One sector can be construed as producing
buildings and infrastructure with very long lifetimes, while the other could rep-
resent the production of machines, transportation equipment, etc., with much
shorter lifetimes. In isolation, each sector produces a self-sustained oscillation
with a period and amplitude determined by the sector's parameter values. How-
ever, when the two sectors are coupled together through their dependence on
each other's output, they tend to synchronize with a rational ratio between the
two periods of oscillation.
The extended long wave model [28] describes the flows of capital plant and
equipment in two capital producing sectors. Each sector uses capital from itself
and from the other sector as the only factors of production. Each sector receives
orders for capital, from itself, from the other sector, and from the consumer
goods sector. Production is made to order (no inventories are kept), and orders
reside in a backlog until capital is produced and delivered.
Each sector i —1,2 maintains a stock Kij of each capital type j = 1,2. The
capital stock is increased by deliveries of new capital and reduced by physical
depreciation. The stock of capital type j depreciates exponentially with and
average lifetime of Tj. The difference in lifetime between the two sectors AT will
be used as a bifurcation parameter to explore the robustness of the aggregated
model.
Output is distributed "fairly" between customers, i.e., the delivery of capital
type j to sector i is the share of total output Xj from sector j , distributed
according to how much sector i has on order with sector j , relative to sector j ' s
total order backlog Bj. Hence,
Kit = x& - ^ (1.3)
Bj Tj
and](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-31-320.jpg)
![Coupled Nonlinear Oscillators 17
S--
Sij = 0{j -Xj-j*-, (1.4)
where a dot denotes time derivative. Sij represents the orders that sector i has
placed with sector j but not yet recieved, and Oij represents the rate of sector
i's new orders for capital from sector j . Each sector receives orders from itself
On, from the other capital sector Oji, and from the consumer goods sector, y,. It
accumulates these orders in a backlog S;, which it then depleted by the sector's
deliveries of capital X{. Hence,
Bi = {on + Oji - y{) - Xi, j ^ i. (1.5)
Production capacity in each sector is determined by a constant-returns-to
scale Cobb-Douglas function of the individual stocks of the two capital types,
with a factor share a £ [0,1] of the other sector's capital type and a share 1 — a
of the sector's own capital type, i.e.,
* = K?K}rKfj, j + i. (1.6)
where the capital-output ratio K, is a constant. The parameter a determines the
degree of coupling between the two sectors. In the simulation studies a is varied
between 0, indicating no interdependence between the sectors, and 1, indicating
the strongest possible coupling where each sector is completely dependent on
capital from the other sector. A characteristic aspect of the Cobb-Douglas
function is that it allows substitution between the two production factors, i.e.,
the same production capacity q can be achieved with different combinations of
Ku and Kij. In this perspective, a is referred to as the elasticity of substitution.
The output Xi from sector i depends on the sector's production capacity Q,
compared to the sector's desired output x*. If desired output is much lower
than capacity, production is cut back, ultimately to zero if no output is desired.
Conversely, if desired output exceeds capacity, output can be increased beyond
capacity, up to a certain limit. In our model, the sector's output is formulated
as
xi = f(^jci (1.7)](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-32-320.jpg)
![18 Chaotic Synchronization: Applications to Living Systems
where the capacity-utilization function /(•) has the form
/(r)=7(l-(l^y), 7>1. (1.8)
With this formulation /(0) = 0, /(l) = 1, and lim^oo /(r) = 7. Thus
the parameter 7 determines the maximum production possible. In the present
analysis we take 7 = 1.1. Note that /(r) > r, r G [0,1], implying that firms
are reluctant to cut back their output when capacity exceeds demand. Instead,
they deplete their backlogs and reduce their delivery delays.
Sector i's desired orders o*j for new capital of type j consist of three compo-
nents. First, all other things being equal, firms will order to replace depreciation
of their existing capital stock, K^/TJ. Second, if their current capital stock is
below (above) its desired level &*• firms will order more (less) capital in order to
correct the discrepancy over time. Third, firms consider the current supply line
Sij of capital and compare it to its desired level s*-. If the supply line is below
(above) that desired, firms order more (less) in order to increase (decrease) the
supply line over time. In total, our expression for the desired ordering rate
becomes
dl.= ^L+KiZ^L + ^ i i (1.9)
where the parameters rf and rf are the characteristic adjustment times for the
capital stock and the supply line, respectively. This decision rule is supported
by extensive empirical [57] and experimental [58, 59] work. The rule is based
on a so-called anchoring-and-adjustment approach that is believed to capture
the bounded rationality of real decision makers.
Actual orders are constrained to be non-zero (cancellation of orders is not
considered) and the fractional rate of expansion of the capital stock is also
assumed to be limited by bottlenecks related to labor, market development,
and other factors not represented in the production function. These constraints
are accounted for through the expression](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-33-320.jpg)
![Coupled Nonlinear Oscillators 19
where orders for new capital are expressed as a factor g(-) times the rate of
capital depreciation. For the function g(-) we have assumed the form
9(u) = r-TT r ^ r (1-11)
where the parameters have the following values /3 = 6, /ii = 27/7, ii — 8/7,
v = 2/3, and 1/2 = 3. These parameters are specified so that #(1) = 1,
g'(l) = 1, and g"(l) = 0. Furthermore limu^00g(u) = f3 and lim^-oo g(u) = 0.
Note that g(u) has a neutral interval around the equilibrium point (u = 1)
where actual orders equal desired orders.
The desired capital stock fc*- is proportional to the desired production rate
x* with a constant capital-output ratio. Thus, it is implicitly assumed that the
relative prices of the two types of capital are constant, so there is no variation
in desired factor proportions. Hence,
tyj ~ K
ijx
i (.I'l^,)
where Ky is the capital-output ratio of capital type j in sector i.
In calculating the desired supply line s*-, firms are assumed to account for
the delivery delay for each type of capital. The target supply line is taken to
be the level at which the deliveries of capital, given the current delivery delay,
would equal the current depreciation of the capital stock. The current delivery
delay of capital from a sector is the sector's backlog divided by its output. Thus,
4 = ^ , (1-13)
Finally, orders from the consumer goods sector j/j are assumed to be exoge-
nous, constant, and equal for both sectors. The last assumption is not without
consequence, since the relative size of the demands for the two types of capital
can change the dynamics of the model considerably [56].
The capital-output ratios and average capital lifetimes are formulated in
such a way that the aggregate equilibrium values of these parameters for the
model economy as a whole remain constant and equal to the values in Sterman's
original model. Specifically, the average capital lifetimes in the two sectors are](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-34-320.jpg)
![Coupled Nonlinear Oscillators 21
of our model is virtually indistinguishable from that of the original model [55].
With the above parameters, the equilibrium point is unstable and the system
quickly settles into a limit cycle with a period of approximately 47 years. Each
new cycle begins with a period of rapid growth, where desired output exceeds
capacity. The capital sector is thereby induced to order more capital, which,
by further swelling order books, fuels the upturn in a self-reinforcing process.
Eventually, capacity catches up with demand, but at this point it far exceeds the
equilibrium level. The self-ordering process is now reversed, as falling orders
from the capital sector lead to falling demand, which collapses to the point
where only the exogenous goods sector places new orders. A long period of
depression follows, during which the excess capital is gradually depleted, until
capacity reaches demand. At this point, the capital sector finally raises enough
orders to offset its own discards, increasing orders above capacity and initiating
the next cycle.
To explore the robustness of the single-sector model to differences in the
parameters governing the individual sectors, we now simulate the model when
some parameters differ between the two sectors. In spite of its simplicity, the
model contains a considerable number of parameters which may differ from
sector to sector. In the present analysis, we vary the difference A T in capital
lifetimes for different values of the coupling parameter a. As described above,
we have scaled all other parameters with the capital-lifetime parameters in
such a way that, when a = 0, each sector is simply a time-scaled version of the
original one-sector model.
Fig 1.8. Synchronization (1:1
mode-locking) in the coupled two-
sector model. The figure shows
the capacity in each of the two
sectors as a function of time in
the steady state. The difference
in capital lifetimes AT is 6 years.
The coupling parameter a is 0.25.
The machinery sector leads the
oscillations.
200 300
Time (Years)](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-36-320.jpg)
![Coupled Nonlinear Oscillators 23
larger excursions in production capacity are found for sector 2 (the "machinery"
sector), which is also the sector that leads in phase.
The lifetime difference of 6 years corresponds to a lifetime for machinery
capital of 17 years and a lifetime of buildings and infrastructure of 23 years. If,
with the same coupling parameter, the difference in capital lifetimes is increased
to Ar=9 years, we observe a doubling of the period. The two sectors now
alternate between high and low maxima for their production capacities. This
type of behavior is referred to as a 2:2 mode. It has developed out of the
synchronous 1:1 mode through a period-doubling bifurcation [60]. The 2:2
solution is illustrated in Fig. 1.9.
As the difference in lifetimes is further increased, the model passes through
a Feigenbaum cascade of period-doubling bifurcations (4:4:, 8:8, etc.) and be-
comes chaotic at approximately AT = 10.4. years. Figure 1.10 shows the chaotic
solution generated when Ar = 10.7 years. Calculation of the largest Lyapunov
exponent confirms that the solution in Fig. 1.10 is chaotic. We conclude that
deterministic chaos can arise in a macroeconomic model that in its aggregated
form supports self-sustained oscillations, if the various sectors (because of dif-
ferences in parameter values) fail to synchronize in a regular motion.
Fig 1.10. Synchronized chaotic
behavior. As the difference
in capital lifetimes is increased
further, the behavior becomes
chaotic. For Ar = 10.7 years,
the model shows irregular behav-
ior, and initial conditions close
to each other quickly diverge.
Nonetheless, the two sectors re-
main locked with a ratio of unity
between their average periods.
A more detailed illustration of the route to chaos is provided by the bifur-
cation diagram in Fig. 1.11. Here we have plotted the maximum production
capacity attained in sector 1 over each cycle as a function of the lifetime differ-
ence Ar. The difference in capital lifetimes spans the interval 6 y < AT < 30 y.
When Ar = 30 y, the lifetime of the short-lived capital is just five years while
the lifetime of the long lived capital is 35 years. The coupling parameter is kept
200 300
Time (Years)](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-38-320.jpg)
![24 Chaotic Synchronization; Applications to Living Systems
Fig 1.11. Bifurcation diagram for
increasing lifetime difference Ar
and constant a. The figure shows
the local maxima attained for the
capacity of the long-lived capital
producer in the steady state for
varying values of the lifetime dif-
ference. From left to right the
main regions of periodic behav-
ior correspond to the 1:1, 1:2, 1:3,
and 1:4 synchronization regions.
constant equal to a = 0.2. Inspection of the figure shows that the 1:1 frequency-
locking, in which the production capacity of sector 1 reaches the same maxi-
mum in each long-wave upswing, is maintained up to A r « 6.4 years, where the
first period-doubling bifurcation occurs. (Identification of the various periodic
modes cannot be made from the bifurcation diagram alone, but involves the
time and phase plots as well.) In the interval 6.4 y < AT < 7.9 y , the long-
wave upswings alternate between a high and a low maximum. Hereafter follows
an interval up to approximately Ar = 8.1 years with 4:4 locking, an interval of
8:8 locking, etc. Within the interval approximately 8.2 y < AT < 11.8 y small
windows of periodic motion are visible between regions of chaos. In the region
around 12.4 y < AT < 13.0 y chaos gives way to the 2:3 mode-locked solution
and the associated period-doubling cascade 4:6, 8:12, etc. Another region of
chaotic behavior follows until about A r « 15.2 years, where the system locks
into 1:2 motion. Similarly the regions of 1:3 and 1:4 entrainment are clearly
visible as A r continues to increase. Note that the 1:4 region bifurcates into 2:8
at around Arft*27.6 years, but then returns to 1:4 motion at A r as 28.3 years,
rather than cascading through further doublings to chaos.
The phase diagram in Fig. 1.12 gives an overview of the dominant modes for
different combinations of the lifetime difference A r and the coupling parameter
or. The zones of mode-locked (i.e., periodic) solutions in this diagram are the
well-known ArnoPd tongues [33, 38, 39]. Besides the 1:1 tongue, the figure shows
a series of l:n tongues, i.e., regions in parameter space where the buildings
industry completes precisely one long-wave oscillation each time the machinery
0.0 -tromtmim n iii,|« prmm, m») |
6 8 10 12 14 16 18 20 22 24 26 28 30
Lifetime difference (years)](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-39-320.jpg)
![26 Chaotic Synchronization: Applications to Living Systems
mechanism in each individual sector. Thus, for high values of a, there is less
competition between the two individual, autonomous oscillations and stronger
synchronization. For large differences in capital lifetimes and low values of the
coupling parameter a, the short-lived capital sector (sector 2) completes several
cycles for each oscillation of the long-lived sector (sector 1). However, as a is
increased, the short-term cycle is reduced in amplitude and, for sufficiently high
values of a, it disappears altogether, resulting in a synchronous 1:1 solution.
The locally stabilizing effect of high values of a creates an interesting distortion
of the Arnol'd tongues in Fig. 1.12. For instance, the figure reveals that both
the 1:1 region and the 2:2 region stretch above and around the other regions
for high values of a.
By employing only a single capital-producing sector, the original long-wave
model [55] represents a simplification of the structure of capital and production.
In reality, "capital" is composed of diverse components with different character-
istics. We have focused on the difference in the average lifetime of capital and
it is clear from our analysis that a disaggregated system with diverse capital
lifetimes exhibits a much wider variety of fluctuations. For moderate differ-
ences in parameters between the sectors, the coupling between sectors has the
effect of merging distinct individual cycles into a more uniform aggregate cycle.
The period of the cycle remains in the 50-year range, although the amplitude
may vary greatly form one cycle to the next. The behavior of the two-sector
model thus retains the essential features of the simple model and is robust to
the aggregation of all firms into a single sector.
Entrainment in the disaggregated model arises only via the coupling intro-
duced by the input-output structure of capital production. Other sources of
coupling were ignored. The most obvious links are created by the price system.
If, for instance, one type of capital is in short supply, one would expect the rel-
ative price of that factor to rise. To the extent that sectors can substitute one
type of capital for another, one would expect demand for the relatively cheaper
capital components to rise. In this way, the price system will cause local imbal-
ances between order and capacity across the sectors to equalize, thus helping to
bring the individual sectors into phase. (We have performed a few preliminary
simulations of a version of the model that includes a price system and these
simulations show an increased tendency for synchronization). The degree of
substitution between capital types in the production function may well be an
important factor: One would expect high elasticities of substitution to yield](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-41-320.jpg)
![Coupled Nonlinear Oscillators 27
stronger synchronization. A next step in the study of coupled economic oscilla-
tors could therefore involve introducing relative prices and differing degrees of
substitution.
Another, more immediate extension of the above discussion would involve
looking at more than two sectors. On the one hand, a wider variety of capital
producers would introduce more variability in the behavior and, hence, less
uniformity. On the other hand, as the system is disaggregated further, the
strength of the individual self-ordering loops is reduced to near zero, and overall
dynamics will more and more be determined by the interaction between sectors.
It would also be interesting to consider the influence of other (more global)
macroeconomic linkages, such as the Keynesian consumption multiplier. Our
preliminary results demonstrate the importance of studying non-linear entrap-
ment in the economy. The intricacies of such phenomena suggest that there is
a vast unexplored domain of research in the area of economic cycles. We sup-
pose that nonlinear interactions could play as large a role in shaping economic
cycles as do the external random shocks on which much of mainstream business
cycle theory relies. At the same time, our discussion points to the similarities
in nature between the problems we meet in macroeconomic systems and in the
biologically oriented problems discribed in other chapters of this book.
Bibliography
[1] J.J. Collins and I. Stewart, Coupled Nonlinear Oscillators and the Symme-
tries of Animal Gaits, J. Nonlinear Science 3, 349-392 (1993).
[2] S.H. Strogatz and I. Stewart, Coupled Oscillators and Biological Synchro-
nization, Scientific American 12, 68-75 (1993).
[3] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal
Concept in Nonlinear Sciences (Cambridge Nonlinear Science Series 12,
Cambridge University Press, 2001).
[4] E. Mosekilde, Topics in Nonlinear Dynamics (World Scientific, Singapore,
1996).](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-42-320.jpg)
![28 Chaotic Synchronization: Applications to Living Systems
[5] C. Hugenii, Horologium Oscillatorium (Parisiis, France, 1673). English
translation: The Pendulum Clock (Iowa State University Press, Ames,
1986).
[6] C. Hayashi, Nonlinear Oscillations in Physical Systems (McGraw-Hill,
New-York, 1964).
[7] I. Blekhman, Synchronization in Science and Technology (ASME Press,
New York, 1988).
[8] R.L. Stratonovich, Topics in the Theory of Random Noise (Gordon and
Breach, New York, 1963).
[9] J.M.T. Thompson and H.B. Stewart, Nonlinear Dynamics and Chaos (Wi-
ley and Sons, Chichester, 1986).
10] L. Glass and M.C. Mackey, From Clocks to Chaos: The Rhythms of Life
(Princeton University Press, Princeton, 1988).
11] G. Leng, Pulsatility in Neuroendocrine Systems (CRC Press, Boca Raton,
1988).
12] Y.-X. Li and A. Goldbeter, Frequency Specificity in Intercellular Commu-
nication, Biophys. J. 55, 125-145 (1989).
13] F. Kronenberg, L.J. Cote, D.M. Linkie, I. Dyrenfurth, and J.A. Downey,
Menopausal Hot Flashes: Thermoregulatory, Cardiovascular, and Circu-
lating Catecholamine and LH Changes, Maturitas 6, 31-43 (1984).
14] F. Kronenberg and R.M. Barnard, Modulation of Menopausal Hot Flashes
by Ambient Temperature, J. Therm. Biol. 17, 43-49 (1992).
15] C. Schafer, M.G. Rosenblum, J. Kurths, and H.-H. Abel, Heartbeat Syn-
chronized with Ventilation, Nature (London) 392, 239-240 (1998).
16] A. Goldbeter (ed.), Cell to Cell Signalling: From Experiments to Theoret-
ical Models (Academic Press, London, 1989).
17] A. Goldbeter and B. Wurster, Regular Oscillations in Suspensions of a
Putatively Chaotic Mutant of Dictostelium Discoideum, Experientia 45,
363-365 (1989).](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-43-320.jpg)
![Coupled Nonlinear Oscillators 29
[18] S. Dan0, P.G. S0rensen, and F. Hynne, Sustained Oscillations in Living
Cells, Nature (London) 402, 320-322 (1999).
[19] N. Kopell, G.B. Ermentrout, M.A. Whittington, and R.D. Traub, Gamma
Rhythms and Beta Rhythms Have Different Synchronization Properties,
Proc. Nat. Acad. Sci. 97, 1867-1872 (2000).
[20] F. Mormann, K. Lehnertz, P. David, and C.E. Elger, Mean Phase Co-
herence as a Measure of Phase Synchronization and Its Application to the
EEG of Epilepsy Patients, Physica D 144, 358-369 (2000).
[21] K. Kaneko, Relevance of Dynamic Clustering to Biological Networks, Phys-
ica D 75, 55-73 (1994).
[22] P.P. Leyssac and L. Baumbach, An Oscillating Intratubular Pressure Re-
sponse to Alterations in Henle Loop Flow in the Rat Kidney, Acta Physiol.
Scand. 117, 415-419 (1983).
[23] K.S. Jensen, E. Mosekilde, and N.-H. Holstein-Rathlou, Self-Sustained Os-
cillations and Chaotic Behaviour in Kidney Pressure Regulation, Mondes
en Develop. 54/55, 91-109 (1986).
[24] N.-H. Holstein-Rathlou, Synchronization of Proximal Intratubular Pressure
Oscillations: Evidence for Interaction between Nephrons, Pfliigers Archiv
408, 438-443 (1987).
[25] H.P. Meissner and M. Preissler, Ionic Mechanisms of the Glucose-Induced
Membrane Potential Changes in /3-Cells, Horm. and Metab. Res. (Suppl.)
10, 91-99 (1980).
[26] E. Gylfe, E. Grapengiesser, and B. Hellman, Propagation of Cytoplasmic
Ca2+
Oscillations in Clusters of Pancreatic B-Cells Exposed to Glucose,
Cell Calcium 12, 229-240 (1991).
[27] J.D. Sterman and E. Mosekilde, Business Cycles and Long Waves: A Be-
havioral Disequilibrium Perspective. In Business Cycles: Theory and Em-
pirical Methods, ed. W. Semmler (Kluwer Academic Publishers, Dordrecht,
1994).
[28] C. Haxholdt, C. Kampmann, E. Mosekilde, and J.D. Sterman, Mode-
Locking and Entrainment of Endogenous Economic Cycles, System Dyn.
Rev. 11, 177-198 (1995).](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-44-320.jpg)
![30 Chaotic Synchronization: Applications to Living Systems
[29] N.F. Rulkov, Images of Synchronized Chaos: Experiments with Circuits,
Chaos 6, 262-279 (1996).
[30] L.M. Pecora, T.L. Carroll, G.A. Johnson, D. J. Mar, and J.F. Heagy, Fun-
damentals of Synchronization in Chaotic Systems, Concepts, and Applica-
tions, Chaos 7, 520-543 (1997).
[31] G. Kolumban, M.P. Kennedy, and L.O. Chua, The Role of Synchronization
in Digital Communications Using Chaos - Part I: Fundamentals of Dig-
ital Communications, IEEE Trans. Circuits and Systems CS-44, 927-935
(1997).
[32] K.M. Short, Steps Toward Unmasking Secure Communications, Int. J. Bi-
furcation and Chaos 4, 959-977 (1994).
[33] V.I. Arnol'd, Small Denominators. I. Mappings of the Circumference onto
Itself, Am. Math. Soc. Transl., Ser. 2, 46, 213-284 (1965).
[34] G.I. Dykman, P.S. Landa, and Yu.I. Neymark, Synchronizing the Chaotic
Oscillations by External Force, Chaos, Solitons and Fractals 1, 339-353
(1991).
[35] V.S. Anishchenko, T.E. Vadivasova, D.E. Postnov, and M.A. Safonova,
Synchronization of Chaos, Int. J. Bifurcation and Chaos 2, 633-644 (1992).
[36] J. Sturis, E. Van Cauter, J.D. Blackman, and K.S. Polonsky, Entrainment
of Pulsatile Insulin Secretion by Oscillatory Glucose Infusion, J. Clin. In-
vest. 87, 439-445 (1991).
[37] M. Bindschadler and J. Sneyd, A Bifurcation Analysis of Two Coupled
Calcium Oscillators, Chaos 11, 237-246 (2001).
[38] M.A. Taylor and I.G. Kevrekidis, Some Common Dynamic Features of
Coupled Reacting Systems, Physica D 51, 274-292 (1991).
[39] C. Knudsen, J. Sturis, and J.S. Thomsen, Generic Bifurcation Structures of
Arnol'd Tongues in Forced Oscillators, Phys. Rev. A 44, 3503-3510 (1991).
[40] M. Rosenblum, A. Pikovsky, J. Kurths, Phase Synchronization of Chaotic
Oscillators, Phys. Rev. Lett. 76, 1804-1807 (1996).](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-45-320.jpg)
![Coupled Nonlinear Oscillators 31
[41] A. Pikovsky, G. Osipov, M. Rosenblum, M. Zaks, and J. Kurths, Attractor-
Repeller Collision and Eyelet Intermittency at the Transition to Phase Syn-
chronization, Phys. Rev. Lett. 79, 47-50 (1997).
[42] V.S. Anishchenko, T.E. Vadivasova, V.V. Astakhov, O.V. Sosnovtseva,
C.W. Wu, and L.O. Chua, Dynamics of Two Coupled Chua's Circuits,
Int. J. Bifurcation and Chaos 5, 1677-1699 (1995).
[43] L. Chua, M. Itoh., L. Kocarev, and K. Eckert, Chaos Synchronization in
Chua's Circuit. In Chua's Circuits: A Paradigmafor Chaos, edited by R.N.
Madan (World Scientific, Singapore, 1993).
[44] V.S. Anishchenko, Dynamical Chaos - Models and Experiments. Appear-
ance Routes and Structure of Chaos in Simple Dynamical Systems (World
Scientific, Singapore, 1995).
[45] O.E. Rossler, An Equation for Continuous Chaos, Phys. Lett. A 57, 397-
398 (1976).
[46] P.A. Samuelson, Interactions Between the Multiplier Analysis and the Prin-
ciple of Acceleration, The Review of Economic Statistics 21, 75-78 (1939).
[47] R. Goodwin, The Nonlinear Accelerator and the Persistence of Business
Cycles, Econometrica 19, 1-17 (1951).
[48] J.W. Forrester, Industrial Dynamics (MIT Press, Cambridge, 1961).
[49] H. Simon, The Sciences of the Artificial (MIT Press, Cambridge, 1969).
[50] D.L. Meadows, Dynamics of Commodity Production Cycles (Wright-Allen
Press, Cambridge, 1970).
[51] Z. Zannitos, The Theory of Oil Tanker Rates (MIT Press, Cambridge,
1966).
[52] V. Zarnowitz, Recent Work on Business Cycles in Historical Perspective:
A Review of Theories and Evidence, The Journal of Economic Literature
23, 523-580 (1985).
[53] J.W. Forrester, Growth Cycles, De Economist 125, 525-543 (1977).](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-46-320.jpg)
![32 Chaotic Synchronization: Applications to Living Systems
[54] E. Mosekilde, E.R. Larsen, J.D. Sterman, and J.S. Thomsen, Nonlinear
Mode-Interaction in the Macroeconomy, Annals of Operations Research
37, 185-215 (1992).
[55] J.D. Sterman, A Behavioral Model of the Economic Long Wave, Journal
of Economic Behavior and Organization 6, 17-53 (1985).
[56] C. E. Kampmann, Disaggregating a Simple Model of the Economic Long
Wave (Working Paper no. D-3641, Sloan School of Management, M.I.T.,
Cambridge, Mass., U.S.A. 1984).
[57] P.M. Senge, A System Dynamics Approach to Investment-Function Formu-
lation and Testing, Socio-Economic Planning Science 14, 269-280 (1980).
[58] J.D. Sterman, Misperceptions of Feedback in Dynamic Decision Mak-
ing, Organizational Behavior and Human Decision Processes 43, 301-335
(1989).
[59] J.D. Sterman, Modeling Managerial Behavior: Misperceptions of Feedback
in a Dynamic Decision Making Experiment, Management Science 35, 321-
339 (1989).
[60] M. Feigenbaum, Quantitative Universality for a Class of Nonlinear Trans-
formations, J. Stat. Phys. 19, 669-706 (1978).](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-47-320.jpg)
![Chapter 2
TRANSVERSE STABILITY OF COUPLED
MAPS
2.1 Riddling, Bubbling, and On-Off Intermittency
In the early 1980's, Fujisaka and Yamada [1] showed how two identical chaotic
systems under variation of the coupling strength can attain a state of full syn-
chronization where the motion of the coupled system takes place on an invariant
subspace of total phase space. In spite of the fact that the systems are chaotic,
their interaction allow them to move precisely in step. For two coupled identical,
one-dimensional maps, for instance, the synchronous motion is one-dimensional
and occurs along the main diagonal in the phase plane. The transverse Lya-
punov exponent Ax provides a measure of the stability of the chaotic attractor
perpendicular to this direction. As long as Ax is negative, a trajectory mov-
ing in the neighborhood of the synchronized chaotic state will, on average, be
attracted towards this state.
Chaotic synchronization has subsequently been studied by a large number
of investigators [2, 3, 4], and a variety of applications for chaos suppression, for
monitoring and control of dynamical systems, and for different communication
purposes have been suggested [5, 6, 7]. Important questions that arise in this
connection concern the form of the basin of attraction for the synchronized
chaotic state and the bifurcations through which this basin, or the attractor
itself, undergoes qualitative changes as a parameter is varied. Under what con-
ditions will the interacting chaotic oscillators be able to synchronize if they are
33](https://image.slidesharecdn.com/68263-250622061924-ec78243a/85/Chaotic-Synchronization-Applications-to-Living-Systems-1st-Edition-Erik-Mosekilde-48-320.jpg)
Chaotic Synchronization Applications to Living Systems 1st Edition Erik Mosekilde
- 1. Chaotic Synchronization Applications to Living Systems 1st Edition Erik Mosekilde pdf download https://ebookgate.com/product/chaotic-synchronization- applications-to-living-systems-1st-edition-erik-mosekilde/ Get Instant Ebook Downloads – Browse at https://ebookgate.com
- 2. Get Your Digital Files Instantly: PDF, ePub, MOBI and More Quick Digital Downloads: PDF, ePub, MOBI and Other Formats An Introduction to Chaotic Dynamical Systems Second Edition Robert Devaney https://ebookgate.com/product/an-introduction-to-chaotic- dynamical-systems-second-edition-robert-devaney/ Synchronization and Arbitration in Digital Systems 1st Edition David J. Kinniment https://ebookgate.com/product/synchronization-and-arbitration-in- digital-systems-1st-edition-david-j-kinniment/ Joint cognitive systems foundations of cognitive systems engineering 1st Edition Erik Hollnagel https://ebookgate.com/product/joint-cognitive-systems- foundations-of-cognitive-systems-engineering-1st-edition-erik- hollnagel/ OLAP Solutions Building Multidimensional Information Systems 2nd Edition Erik Thomsen https://ebookgate.com/product/olap-solutions-building- multidimensional-information-systems-2nd-edition-erik-thomsen/
- 3. Wireless Technologies for Ambient Assisted Living and Healthcare Systems and Applications Premier Reference Source 1st Edition Athina Lazakidou https://ebookgate.com/product/wireless-technologies-for-ambient- assisted-living-and-healthcare-systems-and-applications-premier- reference-source-1st-edition-athina-lazakidou/ Aluminum Alloys Preparation Properties and Applications Preparation Properties and Applications 1st Edition Erik L. Persson https://ebookgate.com/product/aluminum-alloys-preparation- properties-and-applications-preparation-properties-and- applications-1st-edition-erik-l-persson/ Introduction to Mathematical Modeling and Chaotic Dynamics 1st Edition Ranjit Kumar Upadhyay https://ebookgate.com/product/introduction-to-mathematical- modeling-and-chaotic-dynamics-1st-edition-ranjit-kumar-upadhyay/ A First Course In Chaotic Dynamical Systems Theory And Experiment 2nd Edition Robert L. L. Devaney https://ebookgate.com/product/a-first-course-in-chaotic- dynamical-systems-theory-and-experiment-2nd-edition-robert-l-l- devaney/ Robustness and Evolvability in Living Systems Andreas Wagner https://ebookgate.com/product/robustness-and-evolvability-in- living-systems-andreas-wagner/
- 4. WORLD SCIENTIFIC ONLINEAR SCIENC Series Editor: Leon O. Chua HiET HPPLICflTIONS TO LIVING SYSTEMS Erik Mosekilde, Yuri Maistrenko & Dmitry Postnov World Scientific
- 5. CHAOTIC SVNCHRONIZRTIOI APPLICATIONS TO LIVING SYSTEHS
- 6. WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A. MONOGRAPHS AND TREATISES Volume 25: Chaotic Dynamics in Hamiltonian Systems H. Dankowicz Volume 26: Visions of Nonlinear Science in the 21 st Century Edited by J. L Huertas, W.-K. Chen & Ft. N. Madan Volume 27: The Thermomechanics of Nonlinear Irreversible Behaviors — An Introduction G. A. Maugin Volume 28: Applied Nonlinear Dynamics & Chaos of Mechanical Systems with Discontinuities Edited by M. Wiercigroch & B. de Kraker Volume 29: Nonlinear & Parametric Phenomena* V. Damgov Volume 30: Quasi-Conservative Systems: Cycles, Resonances and Chaos A. D. Morozov Volume 31: CNN: A Paradigm for Complexity L. O. Chua Volume 32: From Order to Chaos II L P. Kadanoff Volume 33: Lectures in Synergetics V. I. Sugakov Volume 34: Introduction to Nonlinear Dynamics* L Kocarev & M. P. Kennedy Volume 35: Introduction to Control of Oscillations and Chaos A. L Fradkov & A. Yu. Pogromsky Volume 36: Chaotic Mechanics in Systems with Impacts & Friction B. Blazejczyk-Okolewska, K. Czolczynski, T. Kapitaniak & J. Wojewoda Volume 37: Invariant Sets for Windows — Resonance Structures, Attractors, Fractals and Patterns A. D. Morozov, T. N. Dragunov, S. A. Boykova & O. V. Malysheva Volume 38: Nonlinear Noninteger Order Circuits & Systems — An Introduction P. Arena, R. Caponetto, L Fortuna & D. Porto Volume 39: The Chaos Avant-Garde: Memories of the Early Days of Chaos Theory Edited by Ralph Abraham & Yoshisuke Ueda Volume 40: Advanced Topics in Nonlinear Control Systems Edited by T. P. Leung & H. S. Qin Volume 41: Synchronization in Coupled Chaotic Circuits and Systems C. W. Wu *Forthcoming
- 7. NONLINEAR SCIENCE •* S^WSA vol.** Series Editor; Leon O. Chua Erik Mosekilde The Technical University of Denmark Yuri Maistrenko National Academy of Sciences, Ukraine Dmitry Postnov Saratov state University, Russia World Scientific *awJersey"London'Singapore*HongKong
- 8. Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. CHAOTIC SYNCHRONIZATION Applications to Living Systems Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-02-4789-3 Printed in Singapore by Uto-Print
- 9. PREFACE The cooperative behavior of coupled nonlinear oscillators is of interest in connec- tion with a wide variety of different phenomena in physics, engineering, biology, and economics. Networks of coupled nonlinear oscillators have served as models of spatio-temporal pattern formation and simple forms of turbulence. Systems of coupled nonlinear oscillators may be used to explain how different sectors of the economy adjust their individual commodity cycles relative to one another through the exchange of goods and capital units or via aggregate signals in the form of varying interest rates or raw materials prices. Similarly, in the biolog- ical sciences it is important to understand how a group of cells or functional units, each displaying complicated nonlinear dynamic phenomena, can interact with each other to produce a coordinated response on a higher organizational level. It is well-known, for instance, that waves of synchronized behavior that propagate across the surface of the heart are essential for the muscle cells to act in unison and produce a regular contraction. Waves of synchronized behavior can also be observed to propagate across the insulin producing beta-cells of the pancreas. In many cases the individual oscillators display chaotic dynamics. It has long been recognized, for instance, that the ability of the kidneys to compen- sate for variations in the arterial blood pressure partly rests with controls as- sociated with the individual functional unit (the nephron). The main control is the so-called tubuloglomerular feedback that regulates the incoming blood flow in response to variations in the ionic composition of the fluid leaving the nephron. For rats with normal blood pressure, the individual nephron typically
- 10. vi Preface exhibits regular limit cycle oscillations in the incoming blood flow. For such rats, both in-phase and antiphase synchronization can be observed between ad- jacent nephrons. For spontaneously hypertensive rats, where the pressure vari- ations for the individual nephron are highly irregular, signs of chaotic phase synchronization are observed. In the early 1980's, Fujisaka and Yamada showed how two identical chaotic oscillators under variation of the coupling strength can attain a state of com- plete synchronization in which the motion of the coupled system takes place on an invariant subspace of total phase space. This type of chaotic synchroniza- tion has subsequently been studied by a significant number of investigators, and a variety of applications for chaos suppression, for monitoring and con- trol of dynamical systems, and for different communication purposes have been suggested. Important questions that arise in this connection concern the stability of the synchronized state to noise or to a small parameter mismatch between the interacting oscillators. Other questions relate to the form of the basin of attraction for the synchronized chaotic state and to the bifurcations through which this state loses its stability. Recent studies of these problems have led to the discovery of a large number of new phenomena, including riddled basins of attraction, attractor bubbling, blowout bifurcations, and on-off intermittency. In addition to various electronic systems, synchronization of interacting chaotic oscillators has been observed for laser systems, for coupled supercon- ducting Josephson junctions, and for interacting electrochemical reactors. For systems of three or more coupled oscillators, one can observe the phenomenon of partial synchronization where some of the oscillators synchronize while others do not. This phenomenon is of interest in connection with the development of new types of communication systems where one mixes a message with a chaotic signal. Primarily through the works of Rosenblum and Pikovsky it has become clear that even systems that are quite different in nature (or oscillators that have dif- ferent parameter settings) can exhibit a form of chaotic synchronization where the phases of the interacting oscillators are locked to move in synchrony whereas the amplitudes can develop quite differently. This phenomenon, referred to as chaotic phase synchronization, is of particular importance for living systems where the interacting functional units cannot be assumed to be identical. Kuramoto and Kaneko have initiated the study of clustering in large en-
- 11. Preface vii sembles of interacting chaotic oscillators with a so-called global (i.e., all-to-all) coupling structure. This type of analyses is relevant for instance to economic sectors that interact via the above mentioned aggregate variations in interest rates and raw materials prices. However, biological systems also display many examples of globally coupled oscillators. The beta-cells in the pancreas, for instance, respond to variations in the blood glucose concentration, variations that at least partly are brought about by changes in the cells' aggregate release of insulin. Important questions that arise in this connection relate to the way in which the clusters are formed and break up as the coupling between the oscillators is varied. The purpose of the book is to present and analyze some of the many interest- ing new phenomena that arise in connection with the interaction of two or more chaotic oscillators. Among the subjects that we treat are periodic orbit thresh- old theory, weak stability of chaotic states, and the formation of riddled basins of attraction. In this connection we discuss local and global riddling, the roles of the absorbing and mixed absorbing areas, attractor bubbling, on-off intermit- tency, and the influence of a small parameter mismatch or of an asymmetry in the coupling structure. We also consider partial synchronization, transitions to chaotic phase synchronization, the role of multistability, coherence resonance, and clustering in ensembles of many noise induced oscillators. However, our aim is also to illustrate how all of these concepts can be ap- plied to improve our understanding of systems of interacting biological oscilla- tors. In-phase synchronization, for instance, where the nephrons of the kidney simultaneously perform the same regulatory adjustments of the incoming blood flow, is expected to produce fast and strong overall reactions to a change in the external conditions. In the absence of synchronization, on the other hand, the response of the system in the aggregate is likely to be slower and less pro- nounced. Hence, part of the regulation of the kidney may be associated with transitions between different states of synchronization among the functional units. Besides synchronization of interacting nephrons, the book also discusses chaotic synchronization and riddled basins of attractions for coupled pancre- atic cells, homoclinic transitions to chaotic phase synchronization in coupled microbiological reactors, and clustering in systems of noise excited nerve cells. To a large extent the book is based on contributions that have been made over the last few years by the Chaos Group at the Technical University of Den-
- 12. viii Preface mark, by the Department of Mathematics, the National Academy of Sciences of Ukraine in Kiev, and by the Department of Physics, Saratov State University. We would like to thank our collaborators and students Brian Lading, Alexander Balanov, Tanya Vadivasova, Natasha Janson, Alexey Pavlov, Jacob Laugesen, Alexey Taborov, Vladimir Astakhov, Morten Dines Andersen, Niclas Carlsson, Christian Haxholdt, Christian Kampmann, and Carsten Knudsen for the many contributions they have made to the present work. Arkady Pikovsky, Jiirgen Kurths, Michael Rosenblum, Vladimir Belykh, Igor Belykh, Sergey Kuznetsov, Vadim Anishchenko, Morten Colding-J0rgensen, Jeppe Sturis, John D. Ster- man, Laura Gardini, and Christian Mira are acknowledged for many helpful suggestions. We would also like to thank Niels-Henrik Holstein-Rathlau and Kay-Pong Yip who have made their experimented data on coupled nephrons available to us. Most of all, however, we would like to thank Vladimir Maistrenko, Oleksandr Popovych, Sergiy Yanchuk, and Olga Sosnovtseva who have been our closest collaborators in the study of chaotic synchronization. Without the enthusiastic help from these friend and colleagues, the book would never have been possible. The book has appeared at a time when research in chaotic synchronization is virtually exploding, and new concepts and ideas emerge from week to week. Hence, it is clear that we have not been able to cover all the relevant aspects of the field. We hope that the combination of mathematical theory, model formulation, computer simulations, and experimental results can inspire other researchers in this fascinating area. We have tried to make the book readable to students and young scientists without the highest expertise in chaos theory. On the other hand, the reader is assumed to have a good knowledge about the basic concepts and methods of nonlinear dynamics from previous studies. The book is dedicated to Lis Mosekilde. In her short scientific career she became the internationally most respected Danish expert in the fields of bone remodelling and osteoporosis. Lyngby, November 2001 Erik Mosekilde, Yuri Maistrenko and Dmitry Postnov
- 13. Contents PREFACE v 1 COUPLED NONLINEAR OSCILLATORS 1 1.1 The Role of Synchronization 1 1.2 Synchronization Measures 7 1.3 Mode-Locking of Endogenous Economic Cycles 13 2 TRANSVERSE STABILITY OF COUPLED MAPS 33 2.1 Riddling, Bubbling, and On-Off Intermittency 33 2.2 Weak Stability of the Synchronized Chaotic State 37 2.3 Formation of Riddled Basins of Attraction 41 2.4 Destabilization of Low-Periodic Orbits 44 2.5 Different Riddling Scenarios 49 2.6 Intermingled Basins of Attraction 54 2.7 Partial Synchronization for Three Coupled Maps 56 3 UNFOLDING THE RIDDLING BIFURCATION 75 3.1 Locally and Globally Riddled Basins of Attraction 75 3.2 Conditions for Soft and Hard Riddling 80 3.3 Example of a Soft Riddling Bifurcation 88 3.4 Example of a Hard Riddling Bifurcation 93 3.5 Destabilization Scenario for a — a, 95 3.6 Coupled Intermittency-III Maps 104 3.7 The Contact Bifurcation 109 3.8 Conclusions 116 4 TIME-CONTINUOUS SYSTEMS 123 4.1 Two Coupled Rossler Oscillators 123 4.2 Transverse Destabilization of Low-Periodic Orbits 125 ix
- 14. x Contents 4.3 Riddled Basins 130 4.4 Bifurcation Scenarios for Asynchronous Cycles 134 4.5 The Role of a Small Parameter Mismatch 140 4.6 Influence of Asymmetries in the Coupled System 145 4.7 Transverse Stability of the Equilibrium Point 147 4.8 Partial Synchronization of Coupled Oscillators 154 4.9 Clustering in a System of Four Coupled Oscillators 162 4.10 Arrays of Coupled Rossler Oscillators 166 5 COUPLED PANCREATIC CELLS 177 5.1 The Insulin Producing Beta-Cells 177 5.2 The Bursting Cell Model 181 5.3 Bifurcation Diagrams for the Cell Model 185 5.4 Coupled Chaotically Spiking Cells 192 5.5 Locally Riddled Basins of Attraction 196 5.6 Globally Riddled Basins of Attraction 200 5.7 Effects of Cell Inhomogeneities 203 6 CHAOTIC PHASE SYNCHRONIZATION 211 6.1 Signatures of Phase Synchronization 211 6.2 Bifurcational Analysis 217 6.3 Role of Multistability 222 6.4 Mapping Approach to Multistability 227 6.5 Suppression of the Natural Dynamics 233 6.6 Chaotic Hierarchy in High Dimensions 239 6.7 A Route to High-Order Chaos 249 7 POPULATION DYNAMIC SYSTEMS 259 7.1 A System of Cascaded Microbiological Reactors 259 7.2 The Microbiological Oscillator 262 7.3 Nonautonomous Single-Pool System 265 7.4 Cascaded Two-Pool System 270 7.5 Homoclinic Synchronization Mechanism 274 7.6 One-Dimensional Array of Population Pools 280 7.7 Conclusions 284
- 15. Contents xi 8 CLUSTERING OF GLOBALLY COUPLED MAPS 291 8.1 Ensembles of Coupled Chaotic Oscillators 291 8.2 The Transcritical Riddling Bifurcation 296 8.3 Global Dynamics after a Transcritical Riddling 302 8.4 Riddling and Blowout Scenarios 307 8.5 Influence of a Parameter Mismatch 313 8.6 Stability of tf-Cluster States 318 8.7 Desynchronization of the Coherent Chaotic State 321 8.8 Formation of Nearly Symmetric Clusters 326 8.9 Transverse Stability of Chaotic Clusters 329 8.10 Strongly Asymmetric Two-Cluster Dynamics 334 9 INTERACTING NEPHRONS 349 9.1 Kidney Pressure and Flow Regulation 349 9.2 Single-Nephron Model 354 9.3 Bifurcation Structure of the Single-Nephron Model 359 9.4 Coupled Nephrons 365 9.5 Experimental Results 370 9.6 Phase Multistability 375 9.7 Transition to Synchronous Chaotic Behavior 382 10 COHERENCE RESONANCE OSCILLATORS 395 10.1 But What about the Noise? 395 10.2 Coherence Resonance 400 10.3 Mutual Synchronization 404 10.4 Forced Synchronization 408 10.5 Clustering of Noise-Induced Oscillations 412 INDEX 425
- 16. Chapter 1 COUPLED NONLINEAR OSCILLATORS 1.1 The Role of Synchronization Synchronization occurs when oscillatory (or repetitive) systems via some kind of interaction adjust their behaviors relative to one another so as to attain a state where they work in unison. An essential aspect of many of the games we play as children is to teach us to coordinate our motions. We skip and learn to jump in synchrony with the swinging rope. We run along the beach and learn to avoid the waves that role ashore, and we take dancing lessons to learn to move in step with the music. One of the main problems in the swimming class is to learn to breath in syn- chrony with the strokes. Not necessarily one-to-one, as there are circumstances where it is advantageous to take two (or more) strokes per inhalation. However, the phase relations must be correct if not to drown. In much the same way, a horse has different forms of motion (such as walk, trot, and gallup), and each of these gaits corresponds to a particular rhythm in the movement of its legs [1,2]. At the trotting course, the jockey tries to keep the horse in trot to the highest possible speed. In its free motion, however, a horse is likely to choose the mode that is most comfortable to it (and, perhaps, least energy demanding). As the speed increases the horse will make transitions from walk to trot and from trot to gallup. Synchronization is a universal phenomenon in nonlinear systems [3]. Well- known examples are the synchronization of two (pendulum) clocks hanging on a l
- 17. 2 Chaotic Synchronization: Applications to Living Systems wall, and the synchronization of the moon's rotation with its orbital motion so that the moon always turns the same side towards the earth. A radio receiver functions by synchronizing its internal oscillator with the period of the radio wave so that the difference, i.e. the transmitted signal, can be detected and converted into sound. A microwave emitting diode is placed in a cavity of a spe- cific form and size to make it synchronize with a particular resonance frequency of the cavity. In a previous book [4] we presented results on synchronization of coupled thermostatically controlled radiators and coupled household refrig- erators. Synchronization can also be observed between coupled laser systems and coupled biochemical reactors, and it is clear that one can find thousands of other examples in engineering and physics. At the assembly line one has to ensure an effective synchronization of the various processes for the production to proceed in an efficient manner, and engineers and scientists over and over again exploit the technique of modulating (or chopping) a test signal in order to benefit from the increased sensitivity of phase detection. The history of synchronization dates back at least to Huygens' observations some 300 years ago [5], and both the history and the basic theory are recapitu- lated in a significant number of books and articles [6, 7]. For regular (e.g., limit cycle) oscillators, synchronization implies that the periodicities of the interact- ing systems precisely coincide and that differences in phase remain constant. In the presence of noise (or for chaotic systems) one can weaken the requirements such that the periodicities only have to coincide on average, and the phase differences are allowed to move within certain bounds. One may also accept occasional phase slips, provided that they do not occur too often [8]. One-to-one synchronization is only a simple manifestation of a much more general phenomenon, also known as entrainment, mode locking, or frequency locking. In nonlinear systems, a periodic motion is usually accompanied by a series of harmonics at frequencies of p times the fundamental frequency, where p is an integer. When two nonlinear oscillators interact, mode locking may occur whenever a harmonic frequency of one mode is close to a harmonic of the other. As a result, nonlinear oscillators tend to lock to one another so that one subsystem completes precisely p cycles each time the other subsystem completes q cycles, with p and q as integers [9, 10]. An early experience with this type of phenomenon is the way one excites a swing by forcing it at twice its characteristic frequency, i.e., we move the body through two cycles of a bending and stretching mode for each cycle of the swing. A similar phenomenon is
- 18. Coupled Nonlinear Oscillators 3 utilized (in optics, electronics, etc.) in a wide range of so-called parametric devices. Contrary to the conventional assumption of homeostasis, many physiological systems are unstable and operate in a pulsatile or oscillatory mode [10, 11]. This s the case, for instance, for the production of luteinizing hormone and insulin hat are typically released in two-hour intervals [4]. In several cases it has been >bserved that the cellular response to a pulsatile hormonal signal is stronger han the response to a constant signal of the same magnitude, suggesting that he oscillatory dynamics plays a role in the control of the system [12]. Hormonal elease processes may also become synchronized, and it has been reported, or instance, that the so-called hot flashes that complicate the lives of many vomen during menopause are related to the synchronized release of 5-7 different lormones [13, 14]. The beating of the heart, the respiratory cycle, the circadian rhythm, and the avarian cycle are all examples of more or less regular self-sustained oscillations. The ventilatory signal is clearly visible in spectral analyses of the beat-to-beat variability of the heart signal, and in particular circumstances the two oscillators may lock together so that, for instance, the heart beats three or four times for each respiratory cycle [15]. The jet lag that we experience after a flight to a different time zone is related to the synchronization of our internal (circadian) rhythm to the local day-and-night cycles, and it is often said that women can synchronize their menstrual cycles via specific scents (pheromones) if they live close together. Rhythmic and pulsatile signals are also encountered in intercellular commu- nication [16]. Besides neurons and muscle cells that communicate by trains of electric pulses, examples include the generation of cyclic AMP pulses in slime mold cultures of Dictostelium discoideum [17] and the newly discovered synchronization of the metabolic processes in suspensions of yeast cells [18]. Synchronization of the activity of the muscle cells in the heart is necessary for the cells to act in unison and produce a regular contraction. Similarly, groups of nerve cells must synchronize to produce the characteristic rhythms of the brain or to act as pacemakers for the glands of the hormonal systems [19]. On the other hand, it is well-known that synchronization of the electrical activity of larger groups of cells in the brain plays as essential role in the development of epileptic seizures [20]. However, nonlinear oscillators may also display more complicated forms of
- 19. 4 Chaotic Synchronization: Applications to Living Systems dynamics, and an interesting question that arises over and over again in the bio- logical sciences concerns the collective behavior of a group of cells or functional units that each display strongly nonlinear phenomena [21]. The human kidney, for instance, contains of the order of one million func- tional units, the nephrons. In order to protect its function, the individual nephron disposes of a negative feedback regulation by which it can control the incoming blood flow. However, because of the delay associated with the flow of fluid through the nephron, this regulation tends to be unstable and pro- duce self-sustained oscillations in the various pressures and concentrations [22]. If the arterial blood pressure is high enough, the pressure oscillations in the nephron may become irregular and chaotic [23]. Neighboring nephrons interact with one another through signals that propagate along the afferent arterioles (incoming blood vessels) and, as experiments show, this interaction can lead to a synchronization of the regular pressure oscillations for adjacent nephrons [24]. Fig 1.1. Pressure variations in two neighboring nephrons for a hyperten- sive rat. Note that there is a certain degree of synchronization between the irregular (chaotic) signals. This syn- chronization is found to arise from in- teractions between the nephrons and not from common external influences. '"*" 0 500 1000 Time (sec) It is obviously of interest to examine to what extent similar synchronization phenomena are manifest in the irregular oscillations at higher blood pressures. Figure 1.1 shows an example of the chaotic pressure variations that one can observe (in the proximal tubule) for neighboring nephrons in a hypertensive rat. Although the two signals are strongly irregular, one is tempted to admit that there is a certain degree of synchronization: The most pronounced maxima and minima in the pressure variations occur almost simultaneously. Figure 1.2
- 20. Coupled Nonlinear Oscillators 5 shows a scanning electron microscope picture of the interaction structure for a couple of nephrons. Here, one can see how the common interlobular artery (IA) branches into separate afferent arterioles (af) for the two nephrons. The two ball-formed bundles are the capillary systems (the glomeruli) of the two nephrons. Here, blood constituents like water and salts are filtered into the tubular system of the nephrons and the remaining blood passes out through the efferent arterioles (e/). See Fig. 9.1 for a more detailed sketch of the structure of the nephron. We would like to understand how the interaction between the nephrons in- fluence the overall functioning of the kidney. Will there be circumstances, for instance, where the coupling produces a global synchronization of all the nephrons or will we see the formation of clusters of nephrons in different syn- chronization states? Will transitions between different states of synchronization play a role in the regulation of the kidney or will such transitions be related to the development of particular diseases? Similarly, each of the insulin producing /3-cells of the pancreas exhibits a complicated pattern of oscillations and bursts in its membrane potential [25]. Presumably through their relation to the exchange of calcium between the cell and its surroundings, these bursts control the release of insulin. The /3-cells are arranged in a spiral structure along capillaries and small veins. Via insulin receptors in the cell membrane, each cell can thus react to the release of insulin from cells that are upstream to it. At the same time, the /3-cells are coupled Fig 1.2. Scanning electron microscope pic- ture of the arteriolar system for a couple of adjacent nephrons. The nephrons are as- sumed to interact with one another via mus- cular contractions that propagate along the afferent arterioles (af).
- 21. 6 Chaotic Synchronization: Applications to Living Systems via gap junctions through which ions and small molecules can pass from cell to cell. Again it is of interest to understand how the collective behavior of a group of cells is related to the dynamics of individual cells. Experiments indicate that there will be waves of synchronization moving across larger groups of cells in an islet of Langerhans [26]. In the economic realm, each individual production sector with its charac- teristic capital life time and inventory coverage parameters tends to exhibit an oscillatory response to changes in the external conditions [27]. Overreac- tion, time delays, and reinforcing positive feedback mechanisms may cause the behavior to become destabilized and lead to complicated nonlinear dynamic phenomena. The sectors interact via the exchange of goods and services and via the competition for labor and other resources. A basic problem for the establishment of a dynamic macroeconomic theory is therefore to describe how the various interactions lead to a more or less complete entrainment of the sectors [28]. The problems associated with chaotic synchronization have also attracted a considerable interest in the fields of electronics and radio engineering. Here, the attention centers around the possibilities of developing new types of commu- nication systems that exploit the particular properties of deterministic chaos [29, 30, 31]. Important questions that arise in this connection pertain to the sensitivity of the synchronized state to noise or to a parameter mismatch be- tween the interacting oscillators. Other questions relate to the behavior of the coupled system, once the synchronization breaks down, and to the initial con- ditions for which entrainment can be attained. It is a problem of considerable interest, whether or not one can mask a message by mixing it with a chaotic signal [32]. In order to discuss some of the problems that arise in connection with chaotic synchronization we shall apply a variety of different simple mathematical mod- els. We start in Chapters 2 and 3 by considering a system of two (or three) coupled logistic maps. This leads us to a discussion of the conditions for syn- chronization in systems of coupled Rossler oscillators (Chapter 4) and in a system of two (nearly) identical /3-cells (Chapter 5). In this connection we show that a /3-cell has regions of chaotic dynamics between the different states of periodic bursting. Towards the end of the book the analysis will lead us to consider clustering in systems of many coupled chaotic oscillators (Chapter 8) and to examine interacting coherence resonance (i.e., stochastically excited) os-
- 22. Coupled Nonlinear Oscillators 7 dilators (Chapter 10). On the way we shall discuss the characteristics of chaotic phase synchronization (Chapter 6) and use the obtained results to examine ex- perimental data for the tubular pressure variations in neighboring nephrons (Chapter 9). Let us start, however, by discussing some of the characteristic signatures of synchronization in regular and chaotic systems. Thereafter, we shall use a model of two interacting capital producing sectors of the economy [28] to recall some of the basic concepts of the classical synchronization theory [33] and to illustrate the role of synchronization in macroeconomic systems. 1.2 Synchronization Measures Let us consider some of the phenomena that one can observe in connection with chaotic phase synchronization [34, 35]. This is the type of synchronization that we expect to find between two coupled chaotic oscillators with different param- eters such as, for instance, between neighboring nephrons in a hypertensive rat. The idea is to focus on the similarities between chaotic phase synchronization and the synchronization phenomena we know for regular oscillators. Among of the questions we would like to discuss are: What are the signatures of chaotic phase synchronization? Can we use similar diagnostic tools as we use for regular oscillators? What are the main bifurcation scenarios? First, however, we should perhaps recall some of the characteristics of the synchronization mechanism for regular oscillators [4, 9, 10]. From a mathematical point of view we understand the synchronization of two periodic oscillators as a transition from quasiperiodic motion to regular periodic behavior for the system as a whole. The quasiperiodic behavior is usually described as the motion on a torus. This motion is characterized by the presence of two incommensurate periods, asscociated with the motions of the individual oscillators. As coupling between the oscillators is introduced, both oscillators adjust their motions in response to the motion of the other, and when the coupling becomes strong enough a transition typically occurs where the two periods start to coincide. In the absence of coupling, the phase of each oscillator is a neutrally stable variable. There are no mechanisms that act to correct for a shift in phase. The amplitude, on the other hand, is controlled by a balance between instability and nonlinearity, and dissipation leads to a rapid decay of any pertubation of the amplitude. Hence, we conclude that mutual phase adjustments will be more significant than amplitude modulation [3]
- 23. 8 Chaotic Synchronization: Applications to Living Systems At least for relatively small coupling strengths, the synchronization takes place via a saddle-node bifurcation [33]. On the surface of the torus a stable (node) and an unstable (saddle) cycle simultaneously emerge. Under variation of a control parameter (for instance, a parameter that controls the uncoupled period of one of the oscillators), the two cycles move away from one another along the torus surface to meet again and become annihilated on the opposite side. As a result, the synchronized state exists in a finite range of the control parameter. The typical situations where synchronization occurs are mutually coupled oscillators and periodically forced oscillators. Glass and Mackey [10] have dis- cussed, for instance, how different forms of synchronization can be observed for periodically forced chicken heart cells. Sturis et al. [36] have described how the release of insulin from the pancreas in normal subjects can be synchronized to an external variation in the supply of glucose, and Bindschadler and Sneyd [37] have described how oscillations in the intracellular concentrations of calcium in biological cells can be synchronized via the exchange of ions through the gap junctions. Under such conditions, the relevant control parameters are the frequency mismatch Aw for the uncoupled oscillators and the coupling strength (or forcing amplitude) K. The phenomenon of synchronization can be described from different perspec- tives. If we look at the Fourier spectra of the oscillations, synchronization can be seen as a characteristic evolution of the amplitudes and frequencies of the spectral components. On the other hand, in terms of a phase space analysis, the synchronization mechanisms are the possible ways of transition from an ergodic (or nonresonant) two-dimensional torus (which, as mentioned above, is the image of quasiperiodic behavior) to a limit cycle, being the image of periodic oscillations. Figure 1.3 shows the typical structure of the 1:1 synchronization regime [38, 39]. To be specific, let us talk about the case of forced synchronization. The representative ways on the (Aw, .^-parameter plane are denoted with arrows: Route A: At weak coupling and a relatively small parameter mismatch the onset of resonance on the two-torus corresponds to the crossing of one of the saddle-node bifurcation lines SN. At this point, a pair of limit cycles (a sta- ble and a saddle cycle) emerge on the torus surface. In terms of the Fourier spectra, this transition can be diagnosed from the approach and final merging
- 24. Coupled Nonlinear Oscillators 9 of the spectral peaks corresponding to the forcing signal and the self-sustained oscillations. Route B: In the case of a considerable frequency mismatch, increasing the forcing amplitude leads to the gradual suppression of the self-sustained oscilla- tions of the forced system. The two-torus decreases in size and collapses into a limit cycle. Hence, the synchronized state arises at the curve of torus bi- furcation. In the Fourier spectra, the spectral component of the self-sustained oscillations decreases in amplitude and disappears when the bifurcation line T is reached. B Fig 1.3. Typical structure of 1:1 synchroniza- tion region. SN is a curve of saddle-Eode bifurcation for a pair of stable and saddle limit cycles and SSN is a curve of saddle- node bifurcation for a pair of saddle and unstable limit cycles. BT is the so-called Bogdanov-Takens point and C denotes an- other codimension-2 bifurcation point. T is the torus bifurcation line, and H denotes a line of homoclinic bifurcation. Route C: In the resonant parameter area, increasing the forcing amplitude does not lead to a qualitative change of the stable limit cycle. However, at the curve of saddle-saddle-node bifurcation SSN the saddle limit cycle and an unstable limit cycle (from the inside of the torus) are annihilated. Thus, the invariant torus surface (which is defined by the unstable manifolds of the saddle cycle) no longer exists above this line. This transition cannot be diagnosed by means of Fourier spectra for the synchronous oscillations. Route D: In some (usually narrow) parameter region a specific kind of tran- sition can be detected in which a homoclinic bifurcation plays the key role. "*/V.;- f - i v ! : Aco
- 25. 10 Chaotic Synchronization: Applications to Living Systems Here, one observes a region of bistability, where the stable synchronous solu- tion and the stable nonresonant torus coexist. Under variation of the control parameters, this bistability manifests itself in terms of a hysteretic behavior. Most of the published work on chaotic phase synchronization refers to the case where the chaotic dynamics has appeared through a cascade of period- doubling bifurcations. This type of chaotic dynamics has a characteristic struc- ture which manifests itself both in the rotation of the trajectory around some center point (Fig. 1.4(a)) and in the presence of a characteristic time scale which can be easily measured from the power spectrum (see Fig. 1.4(b)). These fea- tures of period-doubling chaos are important in relation to the problem of phase synchronization because they make it possible to introduce a simple measure of the instantaneous phase of the chaotic oscillations and to consider the mean return time to some Poincare secant as representative of the internal rhythm of the dynamics. (b) 0 0.0 0.5 1.0 1.5 2.0 (0 Fig 1.4. The chaotic attractor of the Rossler model demonstrates the typical features of period doubling chaos. The phase trajectory rotates around some center in the phase space projection (a), and one can see the corresponding peak of the fundamental frequency in the power spectrum (b). A relatively small number of model dynamical systems have served an im- portant role in the investigation of period-doubling chaos. The same is true for the problem of chaotic phase synchronization. Here, we can mention the Rossler model [40, 41], the Chua circuit [42, 43], and the Anishchenko-Astakhov generator [35, 44]. All of these models are three-dimensional oscillators demon- strating the period-doubling route to chaos, with the high-periodic and chaotic
- 26. Coupled Nonlinear Oscillators 11 regimes located around a single unstable equilibrium point of saddle-focus type. During the period-doubling cascade, no additional complexity occurs, and these models thus describe a generic case. By way of example let us base our discussion on the Rossler model [45]: x = —u>y — z, y = UJX + ay, z = (3 + z(x-n), (1.1) where a, (3, and x are control parameters, and UJ defines the characteristic frequency of the oscillations. The chaotic dynamics of this model is well-studied. There are two possible types of chaotic attractor. With i increasing, a cascade of period-doubling bifurcations leads to the emergence of chaos. This is called "spiral chaos". This type of chaos is illustrated in Fig. 1.4(a). If /x increases further, the more complicated chaotic motion referred to as "screw chaos" can be observed [44]. To study the synchronization mechanisms we rewrite the model (1.1) as a nonautonomous system: x = —ujy — z + Ksinujft, y = UJX + ay, z = p + z{x-fi), (1.2) where K is the amplitude of a external harmonic drive, and the forcing fre- quency ojj is fixed at 1.0. As illustrated in Fig. 1.4 (b) there is usually a well developed peak in the power spectrum of period-doubling chaos at a frequency close to the frequency of the period-one limit cycle from which the chaotic dynamics has originated. Following the changes in the peak position, one can find an interval for the con- trol parameter UJ where the peak frequency UJQ is in a rational relation with the frequency of the forcing signal ujf. This method works well both for numerical simulations and for full scale experiments. The first observation of frequency locking for chaotic oscillations was made with a similar approach [34, 35]. Figure 1.5 illustrates the locking of the fundamental frequency of the chaotic oscillations for the nonautonomous Rossler model. For UJ G [0.922; 0.929], a 1:1 locking region is observed. Chaotic oscillations inside and outside this re-
- 27. 12 Chaotic Synchronization: Applications to Living Systems Fig 1.5. Frequency locking region for the chaotic oscillations in the nonautonomous Rossler model with a = 0.2, /? = 0.2, // = 4.0, and K = 0.02. ""'6.920 0.925 0.930 CO gion can be classified as synchronous and asynchronous, respectively. This is confirmed by Fig. 1.6 where the Poincare sections for both cases are plotted. It is interesting to note how Fig. 1.5 reproduces the variation that one ob- serves in connection with the synchroniztion of a regular oscillator, forced by a periodic signal. In the interval of chaotic phase synchronization (0.922 < u> < 0.929), the characteristic frequency of the chaotic oscillations coincides precisely with the forcing frequency. This corresponds to the 1:1 step of the well-know devil's staircase for regular oscillations [4, 9, 10]. On both sides of the synchro- nization interval, the characteristic frequency of the chaotic oscillations shifts away from the forcing frequency. This region of asynchronous chaos replaces the quasiperiodic region for regular oscillators. Inspection of Fig. 1.6(b) shows how the structure of the Poincare section for the asynchronous chaotic dynamics reproduces the characteristic structure of the phase space projection for the autonomous Rossler oscillator (see Fig. 1.4). This implies that our stroboscopic measurements (the Poincare section for a forced system) catch the Rossler oscillator more or less at random in all different positions along its trajectory. In the synchronized chaotic case (Fig. 1.6(a)), on the other hand, although there is a certain scatter in the position of the Rossler system in the stroboscopic map, the system is always found in a relatively narrow region of phase space. Whenever the forcing signal has completed a full period, the synchronized Rossler oscillator is back to nearly the same position in phase space. In the following chapters (particularly in Chapter 6) we shall return to the problem of chaotic phase synchronization to discuss additional signatures of synchronization (e.g., the variation of the Lyapunov exponents) and to examine
- 28. Coupled Nonlinear Oscillators 13 10.0 | (a) 5.0 -5.0 • -10.0 -10.0 -5.0 0.0 5.0 10.0 -10.0 -5.0 0.0 5.0 10.0 X X Fig 1.6. Phase projections of the Poincare section in the nonautonomous Rossler model for synchronous (a) and asynchronous (b) chaos at fj, = 4.0, a = 0.2, /? = 0.2, and K = 0.02. the bifurcations involved in the transition to chaotic phase synchronization. We shall also show that chaotic phase sychronization can develop along the same routes as we have illustrated for regular oscillations in Fig. 1.3. However, let us complete the present chapter with a discussion of the role of synchronization in economic systems. This is an area of research which appears so far to have attracted far too little attention. 1.3 Mode-Locking of Endogenous Economic Cycles Macroeconomic models normally aggregate the individual firms of the economy into sectors with similar products, manufacturing processes and decision rules. Sometimes, only a single sector is considered (e.g., [46, 47]). This simplification is justified on pragmatic grounds by noting that it is impractical to portray sep- arately all the firms in an industry or all the products on the market, and by arguing that the phenomena of interest are captured in sufficient detail by the aggregate formulation [48, 49]. Nevertheless, there are instances where aggre- gation is not justified. Economic sectors associated with the new information and communication technologies, for example, may show rapid growth while other, more traditional sectors show little increase or even decline. Here, dis- aggregation is required to provide a proper perspective of future developments. The oscillatory patterns that one can observe in many economic variables also display widely different periodicities. Commodity cycles in pork prices and slaughter rates typically exhibit a period of 3-4 years while similar cycles for -10.0
- 29. 14 Chaotic Synchronization: Applications to Living Systems chicken and cattle have periodicities of about 30 months and 15 years, re- spectively [50]. These cycles are related to the feeding periods for the various animals. In other sectors, such as the auto industry or the tanker market, one can find periodicities that relate to the lifetime of the capital [4, 51]. Obviously, the various sectors interact with one another. However, the eco- nomic modeling literature is weak in providing guidelines for appropriate aggre- gation of dynamic systems, particularly when there are significant interactions between the individual entities. Models of capital investment, for instance, typ- ically represent the average lead time and lifetime of the plant and equipment used by each firm. In reality there are many types of plant and equipment acquired from many vendors operating with a wide range of lead times. In response to changes in its external conditions, each firm will generate cyclic behaviors whose frequency, damping, and other properties are determined by the parameters characterizing the particular mix of lead times and lifetimes the firm faces. Because the individual firms are coupled to one another via the input-output structure of the economy, each acts as a source of perturbations on the others. How do the different lifetimes and lead times of plant and equipment affect the frequency, phase, amplitude, and coherence of economic cycles? How valid is aggregation of individual firms into single sectors for the purpose of studying macroeconomic fluctuations? Here, the issue of coherence or synchronization becomes important. The economy as a whole experiences aggregate business fluctuations of various frequencies from the short-term business cycle to the long-term Kondratieff cycle [4, 27]. Yet why should the oscillations of the individual firms move in phase so as to produce an aggregate cycle? Given the distribution of parameters among individual firms, why do we observe only a few distinct cycles rather than cycles at all frequencies - cycles which might cancel out at the aggregate level? A common approach to the question of synchronization in economics is to assume that fluctuations in aggregates such as gross domestic product or un- employment arise from external shocks, for example sudden changes in resource supply conditions or variations in fiscal or monetary policy [52]. Forrester [53] suggested instead that synchronization could arise from the endogenous inter- action of multiple nonlinear oscillators, i.e., that the cycles generated by indi- vidual firms become reinforced and entrained with one another. Forrester also proposed that such entrainment could account for the uniqueness of the eco-
- 30. Coupled Nonlinear Oscillators 15 nomic cycles. Oscillatory tendencies of similar periodicity in different parts of the economy would be drawn together to form a subset of distinct modes, such as business cycles, construction cycles, and long waves, and each of these modes would be separated from the next by a wide enough margin to avoid synchro- nization. Until recently, however, these suggestions have not been subjected to rigorous analysis. Roughly speaking, synchronization occurs because the nonlinear structure of the interacting parts of a system creates forces that " nudge" the parts of the sys- tem into phase with one another. As described by Huygens [5], two mechanical clocks, hanging on the same wall, are sometimes observed to synchronize their pendulum movements. Each clock has an escapement mechanisms, a highly nonlinear mechanical devise, that transfers power from the weights to the rod of the pendulum. When a pendulum is close to the position where the escape- ment releases, a small disturbance, such as the faint click from the release of the adjacent clock's escapement, may be enough to trigger the release. Hence, the weak coupling of the clocks, through vibrations in the wall, can bring individual oscillations into phase, provided that the two uncoupled frequences are not too different (see [3] for a more complete discussion of Huygens results). We have previously described how mode-locking and other nonlinear dy- namic phenomena arise in a simple model of the economic long wave [4, 27, 54]. As described by Sterman [55], the model explains the long wave as a self- sustained oscillation arising from instabilities in the ordering and production of capital. An increase in the demand for capital leads to further increases through the investment accelerator or "capital self-ordering", because the ag- gregate capital-producing sector depends on its own output to build up its stock of productive capital. Once a capital expansion gets under way, self-reinforcing processes sustain it beyond its long-term equilibrium, until production catches up with orders. At this point, however, the economy has acquired considerable excess capital, forcing capital production to remain below the level needed for replacement until the excess has been fully depreciated, and room for a new expansion has been created. The concern of the present discussion is the model's aggregation of capital into a single type. The real economy consists of many sectors employing dif- ferent kinds of capital in different amounts. Parameters, such as the average productive life of capital and the relative amounts of different capital compo- nents employed, may vary from sector to sector. In isolation, the buildings
- 31. 16 Chaotic Synchronization: Applications to Living Systems industry may show a temporal variation significantly different from that of, for instance, the machinery industry. An early study by Kampmann [56] took a first step in this direction by disag- gregating the simple long-wave model into a system of several capital producing sectors with different characteristics. Kampmann showed that the multi-sector system could produce a range of different behaviors, at times quite different from the original one-sector model. The present analysis provides a more formal ap- proach, using a two-sector model. One sector can be construed as producing buildings and infrastructure with very long lifetimes, while the other could rep- resent the production of machines, transportation equipment, etc., with much shorter lifetimes. In isolation, each sector produces a self-sustained oscillation with a period and amplitude determined by the sector's parameter values. How- ever, when the two sectors are coupled together through their dependence on each other's output, they tend to synchronize with a rational ratio between the two periods of oscillation. The extended long wave model [28] describes the flows of capital plant and equipment in two capital producing sectors. Each sector uses capital from itself and from the other sector as the only factors of production. Each sector receives orders for capital, from itself, from the other sector, and from the consumer goods sector. Production is made to order (no inventories are kept), and orders reside in a backlog until capital is produced and delivered. Each sector i —1,2 maintains a stock Kij of each capital type j = 1,2. The capital stock is increased by deliveries of new capital and reduced by physical depreciation. The stock of capital type j depreciates exponentially with and average lifetime of Tj. The difference in lifetime between the two sectors AT will be used as a bifurcation parameter to explore the robustness of the aggregated model. Output is distributed "fairly" between customers, i.e., the delivery of capital type j to sector i is the share of total output Xj from sector j , distributed according to how much sector i has on order with sector j , relative to sector j ' s total order backlog Bj. Hence, Kit = x& - ^ (1.3) Bj Tj and
- 32. Coupled Nonlinear Oscillators 17 S-- Sij = 0{j -Xj-j*-, (1.4) where a dot denotes time derivative. Sij represents the orders that sector i has placed with sector j but not yet recieved, and Oij represents the rate of sector i's new orders for capital from sector j . Each sector receives orders from itself On, from the other capital sector Oji, and from the consumer goods sector, y,. It accumulates these orders in a backlog S;, which it then depleted by the sector's deliveries of capital X{. Hence, Bi = {on + Oji - y{) - Xi, j ^ i. (1.5) Production capacity in each sector is determined by a constant-returns-to scale Cobb-Douglas function of the individual stocks of the two capital types, with a factor share a £ [0,1] of the other sector's capital type and a share 1 — a of the sector's own capital type, i.e., * = K?K}rKfj, j + i. (1.6) where the capital-output ratio K, is a constant. The parameter a determines the degree of coupling between the two sectors. In the simulation studies a is varied between 0, indicating no interdependence between the sectors, and 1, indicating the strongest possible coupling where each sector is completely dependent on capital from the other sector. A characteristic aspect of the Cobb-Douglas function is that it allows substitution between the two production factors, i.e., the same production capacity q can be achieved with different combinations of Ku and Kij. In this perspective, a is referred to as the elasticity of substitution. The output Xi from sector i depends on the sector's production capacity Q, compared to the sector's desired output x*. If desired output is much lower than capacity, production is cut back, ultimately to zero if no output is desired. Conversely, if desired output exceeds capacity, output can be increased beyond capacity, up to a certain limit. In our model, the sector's output is formulated as xi = f(^jci (1.7)
- 33. 18 Chaotic Synchronization: Applications to Living Systems where the capacity-utilization function /(•) has the form /(r)=7(l-(l^y), 7>1. (1.8) With this formulation /(0) = 0, /(l) = 1, and lim^oo /(r) = 7. Thus the parameter 7 determines the maximum production possible. In the present analysis we take 7 = 1.1. Note that /(r) > r, r G [0,1], implying that firms are reluctant to cut back their output when capacity exceeds demand. Instead, they deplete their backlogs and reduce their delivery delays. Sector i's desired orders o*j for new capital of type j consist of three compo- nents. First, all other things being equal, firms will order to replace depreciation of their existing capital stock, K^/TJ. Second, if their current capital stock is below (above) its desired level &*• firms will order more (less) capital in order to correct the discrepancy over time. Third, firms consider the current supply line Sij of capital and compare it to its desired level s*-. If the supply line is below (above) that desired, firms order more (less) in order to increase (decrease) the supply line over time. In total, our expression for the desired ordering rate becomes dl.= ^L+KiZ^L + ^ i i (1.9) where the parameters rf and rf are the characteristic adjustment times for the capital stock and the supply line, respectively. This decision rule is supported by extensive empirical [57] and experimental [58, 59] work. The rule is based on a so-called anchoring-and-adjustment approach that is believed to capture the bounded rationality of real decision makers. Actual orders are constrained to be non-zero (cancellation of orders is not considered) and the fractional rate of expansion of the capital stock is also assumed to be limited by bottlenecks related to labor, market development, and other factors not represented in the production function. These constraints are accounted for through the expression
- 34. Coupled Nonlinear Oscillators 19 where orders for new capital are expressed as a factor g(-) times the rate of capital depreciation. For the function g(-) we have assumed the form 9(u) = r-TT r ^ r (1-11) where the parameters have the following values /3 = 6, /ii = 27/7, ii — 8/7, v = 2/3, and 1/2 = 3. These parameters are specified so that #(1) = 1, g'(l) = 1, and g"(l) = 0. Furthermore limu^00g(u) = f3 and lim^-oo g(u) = 0. Note that g(u) has a neutral interval around the equilibrium point (u = 1) where actual orders equal desired orders. The desired capital stock fc*- is proportional to the desired production rate x* with a constant capital-output ratio. Thus, it is implicitly assumed that the relative prices of the two types of capital are constant, so there is no variation in desired factor proportions. Hence, tyj ~ K ijx i (.I'l^,) where Ky is the capital-output ratio of capital type j in sector i. In calculating the desired supply line s*-, firms are assumed to account for the delivery delay for each type of capital. The target supply line is taken to be the level at which the deliveries of capital, given the current delivery delay, would equal the current depreciation of the capital stock. The current delivery delay of capital from a sector is the sector's backlog divided by its output. Thus, 4 = ^ , (1-13) Finally, orders from the consumer goods sector j/j are assumed to be exoge- nous, constant, and equal for both sectors. The last assumption is not without consequence, since the relative size of the demands for the two types of capital can change the dynamics of the model considerably [56]. The capital-output ratios and average capital lifetimes are formulated in such a way that the aggregate equilibrium values of these parameters for the model economy as a whole remain constant and equal to the values in Sterman's original model. Specifically, the average capital lifetimes in the two sectors are
- 35. 20 Chaotic Synchronization: Applications to Living Systems AT , AT TI = T + — a n d T2 = T - — (1.14) and the capital-output ratios are T' T- KU = (l-a)K-, Kij^an-1 , i ^ j , and nt = K}ra Kfj. (1.15) Fig 1.7. Simulation of the one- sector model. The steady-state behavior is a limit cycle with a period of approximately 47 years. The plot shows production ca- pacity, production, and desired production of capital equipment, respectively. All variables are shown on the same scale. Max- ima are reached in the order: De- sired production, actual produc- tion, capacity. The average lifetime of capital r is taken to be 20 years and the average capital- output ratio K = 3 years. The above formulation assures that capacity equals desired output when both capital stocks equal their desired levels and that the equilibrium aggre- gate lifetime of capital and equilibrium aggregate capital-output ratio equal the corresponding original parameters in the one-sector model. Furthermore, parameters in the decision rules were scaled to the average lifetime of capital produced by that sector. Thus, when there is no coupling between the sectors (a = 0), one sector is simply a time-scaled version of the other. Hence, the parameters are r f = r*!*, r f = ^Zl a n d j.= ,jZi (Ll6) T T T where (as in the original model) TK - 1.5 y, r 5 = 1.5 y, and S = 1.5 y. Figure 1.7 shows a simulation of the limit cycle of the one-sector model (a = 0, AT = 0). Even with the modifications we have introduced, the behavior 100 150 Time (Years)
- 36. Coupled Nonlinear Oscillators 21 of our model is virtually indistinguishable from that of the original model [55]. With the above parameters, the equilibrium point is unstable and the system quickly settles into a limit cycle with a period of approximately 47 years. Each new cycle begins with a period of rapid growth, where desired output exceeds capacity. The capital sector is thereby induced to order more capital, which, by further swelling order books, fuels the upturn in a self-reinforcing process. Eventually, capacity catches up with demand, but at this point it far exceeds the equilibrium level. The self-ordering process is now reversed, as falling orders from the capital sector lead to falling demand, which collapses to the point where only the exogenous goods sector places new orders. A long period of depression follows, during which the excess capital is gradually depleted, until capacity reaches demand. At this point, the capital sector finally raises enough orders to offset its own discards, increasing orders above capacity and initiating the next cycle. To explore the robustness of the single-sector model to differences in the parameters governing the individual sectors, we now simulate the model when some parameters differ between the two sectors. In spite of its simplicity, the model contains a considerable number of parameters which may differ from sector to sector. In the present analysis, we vary the difference A T in capital lifetimes for different values of the coupling parameter a. As described above, we have scaled all other parameters with the capital-lifetime parameters in such a way that, when a = 0, each sector is simply a time-scaled version of the original one-sector model. Fig 1.8. Synchronization (1:1 mode-locking) in the coupled two- sector model. The figure shows the capacity in each of the two sectors as a function of time in the steady state. The difference in capital lifetimes AT is 6 years. The coupling parameter a is 0.25. The machinery sector leads the oscillations. 200 300 Time (Years)
- 37. 22 Chaotic Synchronization: Applications to Living Systems In the simulations that follow, sector 1 is always the sector with the longer lifetime of its capital output, corresponding to such industries as housing and infrastructure, while sector 2 has the shorter lifetime parameter, correspond- ing to the machine and equipment sector. Introducing a coupling between the sectors will not only link the behaviors together, but also change the stability properties of the individual sectors. A high value of the coupling parameter a implies that the strength of the capital self-ordering loop in any sector is small. In the extreme case a = 1, each sector will not order any capital from itself. If the delivery delay for capital from the other sector is taken as exogenous and constant, the behavior of an individual sector changes to a highly damped oscil- lation. Indeed, a linear stability analysis around the steady-state equilibrium of an individual sector shows that the equilibrium becomes stable for sufficiently high values of a. As will become evident below, this stability at high values of the coupling parameter has significant effects on the mode-locking behavior of the coupled system. Fig 1.9. 2:2 mode-locking result- ing from a period-doubling bifur- cation. As the difference AT is in- creased to 9 years, the 1:1 mode is replaced by an alternating pat- tern of smaller and larger swings, so that the total period is dou- bled. As in the previous figure, a = 0.25. 0 100 200 300 400 500 Time (Years) As long as the parameters of the two sectors are close enough, we expect synchronization (or 1:1 frequency locking) to occur, i.e., we expect that the different cycles generated by the individual sectors will adjust to one another and exhibit a single aggregate economic long wave with the same period for both sectors. The stronger the coupling a, the stronger the forces of synchronization are expected to be. As an example of such synchronization, Fig. 1.8 shows the outcome of a simulation performed with a difference in capital lifetimes between the two sectors of A T = 6 years and a coupling parameter a = 0.25. The two sectors, although not quite in phase, have identical periods of oscillations. The
- 38. Coupled Nonlinear Oscillators 23 larger excursions in production capacity are found for sector 2 (the "machinery" sector), which is also the sector that leads in phase. The lifetime difference of 6 years corresponds to a lifetime for machinery capital of 17 years and a lifetime of buildings and infrastructure of 23 years. If, with the same coupling parameter, the difference in capital lifetimes is increased to Ar=9 years, we observe a doubling of the period. The two sectors now alternate between high and low maxima for their production capacities. This type of behavior is referred to as a 2:2 mode. It has developed out of the synchronous 1:1 mode through a period-doubling bifurcation [60]. The 2:2 solution is illustrated in Fig. 1.9. As the difference in lifetimes is further increased, the model passes through a Feigenbaum cascade of period-doubling bifurcations (4:4:, 8:8, etc.) and be- comes chaotic at approximately AT = 10.4. years. Figure 1.10 shows the chaotic solution generated when Ar = 10.7 years. Calculation of the largest Lyapunov exponent confirms that the solution in Fig. 1.10 is chaotic. We conclude that deterministic chaos can arise in a macroeconomic model that in its aggregated form supports self-sustained oscillations, if the various sectors (because of dif- ferences in parameter values) fail to synchronize in a regular motion. Fig 1.10. Synchronized chaotic behavior. As the difference in capital lifetimes is increased further, the behavior becomes chaotic. For Ar = 10.7 years, the model shows irregular behav- ior, and initial conditions close to each other quickly diverge. Nonetheless, the two sectors re- main locked with a ratio of unity between their average periods. A more detailed illustration of the route to chaos is provided by the bifur- cation diagram in Fig. 1.11. Here we have plotted the maximum production capacity attained in sector 1 over each cycle as a function of the lifetime differ- ence Ar. The difference in capital lifetimes spans the interval 6 y < AT < 30 y. When Ar = 30 y, the lifetime of the short-lived capital is just five years while the lifetime of the long lived capital is 35 years. The coupling parameter is kept 200 300 Time (Years)
- 39. 24 Chaotic Synchronization; Applications to Living Systems Fig 1.11. Bifurcation diagram for increasing lifetime difference Ar and constant a. The figure shows the local maxima attained for the capacity of the long-lived capital producer in the steady state for varying values of the lifetime dif- ference. From left to right the main regions of periodic behav- ior correspond to the 1:1, 1:2, 1:3, and 1:4 synchronization regions. constant equal to a = 0.2. Inspection of the figure shows that the 1:1 frequency- locking, in which the production capacity of sector 1 reaches the same maxi- mum in each long-wave upswing, is maintained up to A r « 6.4 years, where the first period-doubling bifurcation occurs. (Identification of the various periodic modes cannot be made from the bifurcation diagram alone, but involves the time and phase plots as well.) In the interval 6.4 y < AT < 7.9 y , the long- wave upswings alternate between a high and a low maximum. Hereafter follows an interval up to approximately Ar = 8.1 years with 4:4 locking, an interval of 8:8 locking, etc. Within the interval approximately 8.2 y < AT < 11.8 y small windows of periodic motion are visible between regions of chaos. In the region around 12.4 y < AT < 13.0 y chaos gives way to the 2:3 mode-locked solution and the associated period-doubling cascade 4:6, 8:12, etc. Another region of chaotic behavior follows until about A r « 15.2 years, where the system locks into 1:2 motion. Similarly the regions of 1:3 and 1:4 entrainment are clearly visible as A r continues to increase. Note that the 1:4 region bifurcates into 2:8 at around Arft*27.6 years, but then returns to 1:4 motion at A r as 28.3 years, rather than cascading through further doublings to chaos. The phase diagram in Fig. 1.12 gives an overview of the dominant modes for different combinations of the lifetime difference A r and the coupling parameter or. The zones of mode-locked (i.e., periodic) solutions in this diagram are the well-known ArnoPd tongues [33, 38, 39]. Besides the 1:1 tongue, the figure shows a series of l:n tongues, i.e., regions in parameter space where the buildings industry completes precisely one long-wave oscillation each time the machinery 0.0 -tromtmim n iii,|« prmm, m») | 6 8 10 12 14 16 18 20 22 24 26 28 30 Lifetime difference (years)
- 40. Coupled Nonlinear Oscillators 25 industry completes n oscillations. Between these tongues, regions with other commensurate wave periods may be observed. An example is the 2:3 tongue found in the area around a = 0.15 and Ar = 12 years. Similar to the 2:2 period-doubled solution on the right-hand side of the 1:1 tongue, there is a 2:4 period-doubled solution along part of the right-hand edge of the 1:2 tongue. 1.0 0.9 0.8 0.7 0.6 SacMr I Ua*tnt4 Seao. 2 Mori-livid capital produce) it ruble capital producer) • ruble la ieiaUioa hr alpha, ta kraWoa Ac aloha abova Hue too. IIH|llllimi|lllllllll|lllllllll|lllllllll|lllllllll|lllllllll|Tllllllll|lllllllll|lllllllll 12 14 16 18 20 22 24 26 28 30 Lifetime difference (Years) Fig 1.12. Parameter phase diagram. The figure summarizes the steady-state behavior of the two-sector model for different combinations of the coupling parameter a and the lifetime differ- ence Ar. A region labeled p : q indicates the area in parameter space where the model shows periodic mode-locked behavior of p cycles for sector 1 and q cycles for sector 2. The dashed curves across the diagram indicate the value of a above which each sector in isolation becomes stable. Above these lines synchronous 1:1 behavior prevails. The phase diagram in Fig. 1.12 also reveals that the synchronous 1:1 so- lution extends to the full range of the lifetime differences Ar for sufficiently high values of the coupling parameter a. When a is large enough, the indi- vidual sectors become stable, if the delivery delay and demand from the other sector are taken as exogenous. For reference, two curves have been drawn in Fig. 1.12, defining the regions in which one or both of the individual sectors are stable. As a increases, the overall behavior is increasingly derived from the cou- pling between the sectors and less and less from the autonomous self-ordering
- 41. 26 Chaotic Synchronization: Applications to Living Systems mechanism in each individual sector. Thus, for high values of a, there is less competition between the two individual, autonomous oscillations and stronger synchronization. For large differences in capital lifetimes and low values of the coupling parameter a, the short-lived capital sector (sector 2) completes several cycles for each oscillation of the long-lived sector (sector 1). However, as a is increased, the short-term cycle is reduced in amplitude and, for sufficiently high values of a, it disappears altogether, resulting in a synchronous 1:1 solution. The locally stabilizing effect of high values of a creates an interesting distortion of the Arnol'd tongues in Fig. 1.12. For instance, the figure reveals that both the 1:1 region and the 2:2 region stretch above and around the other regions for high values of a. By employing only a single capital-producing sector, the original long-wave model [55] represents a simplification of the structure of capital and production. In reality, "capital" is composed of diverse components with different character- istics. We have focused on the difference in the average lifetime of capital and it is clear from our analysis that a disaggregated system with diverse capital lifetimes exhibits a much wider variety of fluctuations. For moderate differ- ences in parameters between the sectors, the coupling between sectors has the effect of merging distinct individual cycles into a more uniform aggregate cycle. The period of the cycle remains in the 50-year range, although the amplitude may vary greatly form one cycle to the next. The behavior of the two-sector model thus retains the essential features of the simple model and is robust to the aggregation of all firms into a single sector. Entrainment in the disaggregated model arises only via the coupling intro- duced by the input-output structure of capital production. Other sources of coupling were ignored. The most obvious links are created by the price system. If, for instance, one type of capital is in short supply, one would expect the rel- ative price of that factor to rise. To the extent that sectors can substitute one type of capital for another, one would expect demand for the relatively cheaper capital components to rise. In this way, the price system will cause local imbal- ances between order and capacity across the sectors to equalize, thus helping to bring the individual sectors into phase. (We have performed a few preliminary simulations of a version of the model that includes a price system and these simulations show an increased tendency for synchronization). The degree of substitution between capital types in the production function may well be an important factor: One would expect high elasticities of substitution to yield
- 42. Coupled Nonlinear Oscillators 27 stronger synchronization. A next step in the study of coupled economic oscilla- tors could therefore involve introducing relative prices and differing degrees of substitution. Another, more immediate extension of the above discussion would involve looking at more than two sectors. On the one hand, a wider variety of capital producers would introduce more variability in the behavior and, hence, less uniformity. On the other hand, as the system is disaggregated further, the strength of the individual self-ordering loops is reduced to near zero, and overall dynamics will more and more be determined by the interaction between sectors. It would also be interesting to consider the influence of other (more global) macroeconomic linkages, such as the Keynesian consumption multiplier. Our preliminary results demonstrate the importance of studying non-linear entrap- ment in the economy. The intricacies of such phenomena suggest that there is a vast unexplored domain of research in the area of economic cycles. We sup- pose that nonlinear interactions could play as large a role in shaping economic cycles as do the external random shocks on which much of mainstream business cycle theory relies. At the same time, our discussion points to the similarities in nature between the problems we meet in macroeconomic systems and in the biologically oriented problems discribed in other chapters of this book. Bibliography [1] J.J. Collins and I. Stewart, Coupled Nonlinear Oscillators and the Symme- tries of Animal Gaits, J. Nonlinear Science 3, 349-392 (1993). [2] S.H. Strogatz and I. Stewart, Coupled Oscillators and Biological Synchro- nization, Scientific American 12, 68-75 (1993). [3] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge Nonlinear Science Series 12, Cambridge University Press, 2001). [4] E. Mosekilde, Topics in Nonlinear Dynamics (World Scientific, Singapore, 1996).
- 43. 28 Chaotic Synchronization: Applications to Living Systems [5] C. Hugenii, Horologium Oscillatorium (Parisiis, France, 1673). English translation: The Pendulum Clock (Iowa State University Press, Ames, 1986). [6] C. Hayashi, Nonlinear Oscillations in Physical Systems (McGraw-Hill, New-York, 1964). [7] I. Blekhman, Synchronization in Science and Technology (ASME Press, New York, 1988). [8] R.L. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1963). [9] J.M.T. Thompson and H.B. Stewart, Nonlinear Dynamics and Chaos (Wi- ley and Sons, Chichester, 1986). 10] L. Glass and M.C. Mackey, From Clocks to Chaos: The Rhythms of Life (Princeton University Press, Princeton, 1988). 11] G. Leng, Pulsatility in Neuroendocrine Systems (CRC Press, Boca Raton, 1988). 12] Y.-X. Li and A. Goldbeter, Frequency Specificity in Intercellular Commu- nication, Biophys. J. 55, 125-145 (1989). 13] F. Kronenberg, L.J. Cote, D.M. Linkie, I. Dyrenfurth, and J.A. Downey, Menopausal Hot Flashes: Thermoregulatory, Cardiovascular, and Circu- lating Catecholamine and LH Changes, Maturitas 6, 31-43 (1984). 14] F. Kronenberg and R.M. Barnard, Modulation of Menopausal Hot Flashes by Ambient Temperature, J. Therm. Biol. 17, 43-49 (1992). 15] C. Schafer, M.G. Rosenblum, J. Kurths, and H.-H. Abel, Heartbeat Syn- chronized with Ventilation, Nature (London) 392, 239-240 (1998). 16] A. Goldbeter (ed.), Cell to Cell Signalling: From Experiments to Theoret- ical Models (Academic Press, London, 1989). 17] A. Goldbeter and B. Wurster, Regular Oscillations in Suspensions of a Putatively Chaotic Mutant of Dictostelium Discoideum, Experientia 45, 363-365 (1989).
- 44. Coupled Nonlinear Oscillators 29 [18] S. Dan0, P.G. S0rensen, and F. Hynne, Sustained Oscillations in Living Cells, Nature (London) 402, 320-322 (1999). [19] N. Kopell, G.B. Ermentrout, M.A. Whittington, and R.D. Traub, Gamma Rhythms and Beta Rhythms Have Different Synchronization Properties, Proc. Nat. Acad. Sci. 97, 1867-1872 (2000). [20] F. Mormann, K. Lehnertz, P. David, and C.E. Elger, Mean Phase Co- herence as a Measure of Phase Synchronization and Its Application to the EEG of Epilepsy Patients, Physica D 144, 358-369 (2000). [21] K. Kaneko, Relevance of Dynamic Clustering to Biological Networks, Phys- ica D 75, 55-73 (1994). [22] P.P. Leyssac and L. Baumbach, An Oscillating Intratubular Pressure Re- sponse to Alterations in Henle Loop Flow in the Rat Kidney, Acta Physiol. Scand. 117, 415-419 (1983). [23] K.S. Jensen, E. Mosekilde, and N.-H. Holstein-Rathlou, Self-Sustained Os- cillations and Chaotic Behaviour in Kidney Pressure Regulation, Mondes en Develop. 54/55, 91-109 (1986). [24] N.-H. Holstein-Rathlou, Synchronization of Proximal Intratubular Pressure Oscillations: Evidence for Interaction between Nephrons, Pfliigers Archiv 408, 438-443 (1987). [25] H.P. Meissner and M. Preissler, Ionic Mechanisms of the Glucose-Induced Membrane Potential Changes in /3-Cells, Horm. and Metab. Res. (Suppl.) 10, 91-99 (1980). [26] E. Gylfe, E. Grapengiesser, and B. Hellman, Propagation of Cytoplasmic Ca2+ Oscillations in Clusters of Pancreatic B-Cells Exposed to Glucose, Cell Calcium 12, 229-240 (1991). [27] J.D. Sterman and E. Mosekilde, Business Cycles and Long Waves: A Be- havioral Disequilibrium Perspective. In Business Cycles: Theory and Em- pirical Methods, ed. W. Semmler (Kluwer Academic Publishers, Dordrecht, 1994). [28] C. Haxholdt, C. Kampmann, E. Mosekilde, and J.D. Sterman, Mode- Locking and Entrainment of Endogenous Economic Cycles, System Dyn. Rev. 11, 177-198 (1995).
- 45. 30 Chaotic Synchronization: Applications to Living Systems [29] N.F. Rulkov, Images of Synchronized Chaos: Experiments with Circuits, Chaos 6, 262-279 (1996). [30] L.M. Pecora, T.L. Carroll, G.A. Johnson, D. J. Mar, and J.F. Heagy, Fun- damentals of Synchronization in Chaotic Systems, Concepts, and Applica- tions, Chaos 7, 520-543 (1997). [31] G. Kolumban, M.P. Kennedy, and L.O. Chua, The Role of Synchronization in Digital Communications Using Chaos - Part I: Fundamentals of Dig- ital Communications, IEEE Trans. Circuits and Systems CS-44, 927-935 (1997). [32] K.M. Short, Steps Toward Unmasking Secure Communications, Int. J. Bi- furcation and Chaos 4, 959-977 (1994). [33] V.I. Arnol'd, Small Denominators. I. Mappings of the Circumference onto Itself, Am. Math. Soc. Transl., Ser. 2, 46, 213-284 (1965). [34] G.I. Dykman, P.S. Landa, and Yu.I. Neymark, Synchronizing the Chaotic Oscillations by External Force, Chaos, Solitons and Fractals 1, 339-353 (1991). [35] V.S. Anishchenko, T.E. Vadivasova, D.E. Postnov, and M.A. Safonova, Synchronization of Chaos, Int. J. Bifurcation and Chaos 2, 633-644 (1992). [36] J. Sturis, E. Van Cauter, J.D. Blackman, and K.S. Polonsky, Entrainment of Pulsatile Insulin Secretion by Oscillatory Glucose Infusion, J. Clin. In- vest. 87, 439-445 (1991). [37] M. Bindschadler and J. Sneyd, A Bifurcation Analysis of Two Coupled Calcium Oscillators, Chaos 11, 237-246 (2001). [38] M.A. Taylor and I.G. Kevrekidis, Some Common Dynamic Features of Coupled Reacting Systems, Physica D 51, 274-292 (1991). [39] C. Knudsen, J. Sturis, and J.S. Thomsen, Generic Bifurcation Structures of Arnol'd Tongues in Forced Oscillators, Phys. Rev. A 44, 3503-3510 (1991). [40] M. Rosenblum, A. Pikovsky, J. Kurths, Phase Synchronization of Chaotic Oscillators, Phys. Rev. Lett. 76, 1804-1807 (1996).
- 46. Coupled Nonlinear Oscillators 31 [41] A. Pikovsky, G. Osipov, M. Rosenblum, M. Zaks, and J. Kurths, Attractor- Repeller Collision and Eyelet Intermittency at the Transition to Phase Syn- chronization, Phys. Rev. Lett. 79, 47-50 (1997). [42] V.S. Anishchenko, T.E. Vadivasova, V.V. Astakhov, O.V. Sosnovtseva, C.W. Wu, and L.O. Chua, Dynamics of Two Coupled Chua's Circuits, Int. J. Bifurcation and Chaos 5, 1677-1699 (1995). [43] L. Chua, M. Itoh., L. Kocarev, and K. Eckert, Chaos Synchronization in Chua's Circuit. In Chua's Circuits: A Paradigmafor Chaos, edited by R.N. Madan (World Scientific, Singapore, 1993). [44] V.S. Anishchenko, Dynamical Chaos - Models and Experiments. Appear- ance Routes and Structure of Chaos in Simple Dynamical Systems (World Scientific, Singapore, 1995). [45] O.E. Rossler, An Equation for Continuous Chaos, Phys. Lett. A 57, 397- 398 (1976). [46] P.A. Samuelson, Interactions Between the Multiplier Analysis and the Prin- ciple of Acceleration, The Review of Economic Statistics 21, 75-78 (1939). [47] R. Goodwin, The Nonlinear Accelerator and the Persistence of Business Cycles, Econometrica 19, 1-17 (1951). [48] J.W. Forrester, Industrial Dynamics (MIT Press, Cambridge, 1961). [49] H. Simon, The Sciences of the Artificial (MIT Press, Cambridge, 1969). [50] D.L. Meadows, Dynamics of Commodity Production Cycles (Wright-Allen Press, Cambridge, 1970). [51] Z. Zannitos, The Theory of Oil Tanker Rates (MIT Press, Cambridge, 1966). [52] V. Zarnowitz, Recent Work on Business Cycles in Historical Perspective: A Review of Theories and Evidence, The Journal of Economic Literature 23, 523-580 (1985). [53] J.W. Forrester, Growth Cycles, De Economist 125, 525-543 (1977).
- 47. 32 Chaotic Synchronization: Applications to Living Systems [54] E. Mosekilde, E.R. Larsen, J.D. Sterman, and J.S. Thomsen, Nonlinear Mode-Interaction in the Macroeconomy, Annals of Operations Research 37, 185-215 (1992). [55] J.D. Sterman, A Behavioral Model of the Economic Long Wave, Journal of Economic Behavior and Organization 6, 17-53 (1985). [56] C. E. Kampmann, Disaggregating a Simple Model of the Economic Long Wave (Working Paper no. D-3641, Sloan School of Management, M.I.T., Cambridge, Mass., U.S.A. 1984). [57] P.M. Senge, A System Dynamics Approach to Investment-Function Formu- lation and Testing, Socio-Economic Planning Science 14, 269-280 (1980). [58] J.D. Sterman, Misperceptions of Feedback in Dynamic Decision Mak- ing, Organizational Behavior and Human Decision Processes 43, 301-335 (1989). [59] J.D. Sterman, Modeling Managerial Behavior: Misperceptions of Feedback in a Dynamic Decision Making Experiment, Management Science 35, 321- 339 (1989). [60] M. Feigenbaum, Quantitative Universality for a Class of Nonlinear Trans- formations, J. Stat. Phys. 19, 669-706 (1978).
- 48. Chapter 2 TRANSVERSE STABILITY OF COUPLED MAPS 2.1 Riddling, Bubbling, and On-Off Intermittency In the early 1980's, Fujisaka and Yamada [1] showed how two identical chaotic systems under variation of the coupling strength can attain a state of full syn- chronization where the motion of the coupled system takes place on an invariant subspace of total phase space. In spite of the fact that the systems are chaotic, their interaction allow them to move precisely in step. For two coupled identical, one-dimensional maps, for instance, the synchronous motion is one-dimensional and occurs along the main diagonal in the phase plane. The transverse Lya- punov exponent Ax provides a measure of the stability of the chaotic attractor perpendicular to this direction. As long as Ax is negative, a trajectory mov- ing in the neighborhood of the synchronized chaotic state will, on average, be attracted towards this state. Chaotic synchronization has subsequently been studied by a large number of investigators [2, 3, 4], and a variety of applications for chaos suppression, for monitoring and control of dynamical systems, and for different communication purposes have been suggested [5, 6, 7]. Important questions that arise in this connection concern the form of the basin of attraction for the synchronized chaotic state and the bifurcations through which this basin, or the attractor itself, undergoes qualitative changes as a parameter is varied. Under what con- ditions will the interacting chaotic oscillators be able to synchronize if they are 33
- 49. Discovering Diverse Content Through Random Scribd Documents
- 50. for them. If he went to the pharmacy in Hudson Street he would be back in half an hour. "All right, dad. I know the way. I'm an old hand in New York by this time." He was at the door when Quidmore called him back. "Say, boy. Give us a kiss." Tom was stupefied. He had kissed his adopted mother often enough, but he had never been asked to do this. Quidmore laughed, pulling him close. "Ah, come along! I don't ask you often. You're a fine boy, Tom. You must know as well as I do what's been...." The words were suspended by a hug; but once he was free Tom fled away like a small young wild thing, released from human hands. Having reached the street, he began to feel frightened, prescient, awed. Something was going to happen, he could not imagine what. He made his purchases hurriedly, and then delayed his return. He could be tender with the man; he could be loving; but he couldn't share his secrets. But he had to go back. In the dim upper hall outside the door he paused to pump up courage to go in. He was not afraid in the common way of fear; he was only overcome with apprehension at having a knowledge he rejected forced on him. The first thing he noticed was that no light came through the crack beneath the door. The room was apparently dark. That was strange because his father dreaded darkness, except when he was there to keep him company. He crept to the door and listened. There was no sound. He pushed the door open. The lights were out. In panic at what he might discover, he switched on the electricity. But he only found the room empty. That was so far a relief. His father had gone out, and would be back again. Closing the door behind him, he advanced into the room.
- 51. It seemed more than empty. It felt abandoned, as if something had gone which would not return. He remembered that sensation afterward. He stood still to wonder, to conjecture. The Red Indian gleamed with his bronze leer. The next thing the boy noticed was an odd little pile on the table. It was money—notes. On top of the notes there was silver and copper. He stooped over them, touching them with his forefinger, pushing them. He pushed them as he might have pushed an insect to see whether or not it was alive. Lastly he noticed a paper, on which the money had been placed. There was something scribbled on it with a pencil. He held it under the dim lamp. "For Tom—with a real love." The tears gushed to his eyes, as they always did when people showed that they loved him. But he didn't actually cry; he only stood still and wondered. He couldn't make it out. That his father should have gone out and forgotten all his money was unusual enough, but that he should have left these penciled words was puzzling. It was easy to count the money. There were seven fifty-dollar bills, with twenty-eight dollars and fifty-four cents in smaller bills and change. He seemed to remember that his father had drawn four hundred dollars for the Wilmington expenses, with a margin for purchases. He stood wondering. He could never recall how long he stood wondering. The rest of the night became more or less a blank to him; for, to the best of the boy's knowledge, the man who had adopted him was never seen again. XX TO the best of the boy's knowledge the man who had adopted him was never seen again; but it took some time to assume the fact that he was dead. Visitors to New York often dived below the surface, to
- 52. come up again a week or ten days later. Their experience in these absences they were not always eager to discuss. "Why, I've knowed 'em to stay away that long as yer'd swear they'd been kidnapped," Mr. Honeybun informed the boy. "He's on a little time; that's all. Nothink but nat'rel to a man of his age—and a widower—livin' in the country—when he gits a bit of freedom in the city." "Yes, but what'll he do for money?" There was this point of view, to be sure. Mr. Goodsir suggested that Quidmore had had more money still, that he had only left this sum to cover Tom's expenses while he was away. "And listen, son," he continued, kindly, "that's a terr'ble big wad for a boy like you to wear on his person. Why, there's guys that free- quents this very house that'd rob and murder you for half as much, and never drop a tear. Now here I am, an old trusty man, accustomed to handle funds, and not sneak nothin' for myself. If I could be of any use to you in takin' charge of it like...." "Me and you'll talk this over, later," Mr. Honeybun intervened, tactfully. "The kid don't need no one to take care of his cash when his father may skin home again before to-night. Let's wait a bit. If he's goin' to trust anybody it'll be us, his next of kin in this 'ere 'ouse, of course. That'd be so, kiddy, wouldn't it?" Tom replied that it would be so, giving them to understand that he counted on their good offices. For the present he was keeping himself in the non-committal attitude natural to suspense. "You see," he explained, looking from one to another, with his engaging candor, "I can't do anything but just wait and see if he's coming back again, at any rate, not for a spell." The worthies going to their work, the interview ended. At least, Mr. Goodsir went to his work, though within a few minutes Mr. Honeybun was back in Tom's room again.
- 53. "Say, kid; don't you let them three hundred bucks out'n yer own 'and. I can't stop now; but when I blow in to eat at noon I'll tell yer what I'd do with 'em, if you was me. Keep 'em buttoned up in yer inside pocket; and don't 'ang round in this old hut any more'n you can help till I come back and git you. Yer never knows who's on the same floor with yer; but out in the street yer'll be safe." Out in the street he kept to the more populous thoroughfares, coasting the line of docks especially. He liked them. On the façades of the low buildings he could read names which distilled romance into syllables—New Orleans, Savannah, Galveston, Texas, Arizona, Oklahoma. He had always been fond of geography. It opened up the world. It told of countries and cities he would one day visit, and which in the meantime he could dream about. Over the low roofs of the dock buildings he could see the tops of funnels. Here and there was the long black flank of a steamer at its pier. There were flags flying from one masthead or another, while exotic seafaring types slipped in and out amid the crush of vehicles, or dodged the freight train aimlessly shunting up and down. The movement and color, the rumble of deep sound, the confused world-wide purpose of it all, the knowledge that he himself was so insignificant a figure that no robber or murderer would suspect that he had all that money buttoned against his breast, dulled his mind to his desolation. He tried to keep moving so as to make it seem to a suspicious populace that he was an errand boy; but now and then the sense of his loneliness smote him to a standstill. He would wonder where he was going, and what he was going for, as he wondered the same thing about the steamer on the Hudson. Like her, he seemed to be afloat. She, of course, had her destination; but he had nothing in the world to tie up to. He seemed to have heard of a ship that was always sailing—sailing—sailing—sailing—with never a port to have come out of, and never a port in view, The Church of the Sea! He read the words on the corner of a big white building where Jane Street flows toward the docks. He read them again. He read them
- 54. because he liked their suggestions—immensity, solitude, danger perhaps, and God! "THAT'S A TERR'BLE BIG WAD FOR A BOY LIKE YOU TO WEAR" It was queer to think of God being out there, where there were only waves and ships and sailors, but chiefly waves and a few seabirds. It recalled the religion of crippled Bertie Tollivant, the cynic. To the instructed like himself, God was in the churches that had steeples and pews and strawberry sociables, or in the parlors where they held family prayers. They told you that He was everywhere; but that only meant that you couldn't do wrong, you couldn't swear, or smoke a cigarette, or upset some householder's ash-barrels, without His spotting you. Tom Quidmore did not believe that Mr. and Mrs. Tollivant would have sanctioned this Church of the Sea, where God was as free as wind, and over you like the sky, and beyond any human power to monopolize or give away. It made Him too close at hand, too easy to find, and probably much too tender toward sailors, who were often drunk, and homeless little boys. He turned away from the Church of the Sea, secretly envying Bertie Tollivant his
- 55. graceless creed, but not daring to question the wisdom of adult men and women. By the steps of the chop saloon he waited for Mr. Honeybun, who came swinging along, a strong and supple figure, a little after the whistle blew at twelve. To the boy's imagination, now that he had been informed as to his friend's status, he looked like what had been defined to him as a socialist. That is, he had the sort of sinuosity that could slip through half-open windows, or wriggle in at coal- holes, or glide noiselessly up and down staircases. It was ridiculous to say it of one so bony and powerful, but the spring of his step was spiritlike. "Good for you, lad, to be waitin'! We'll go right along and do it, and then it'll be off our minds." What "it" was to be, Tom had no idea. But then he had no suspicions. In spite of his hard childhood, it did not occur to him that grown-up men would do him wrong. He had no fear of Mr. Honeybun, and no mistrust, not any more than a baby in arms has fear or mistrust of its nurse. "And there's another thing," Mr. Honeybun brought up, as they went along. "It don't seem to me no good for a husky boy like you to be just doin' nothink, even while he's waitin' for his pop. I'd git a job, if you was me." The boy said that he would gladly have a job, but didn't know how to get one. "I've got one for yer if yer'll take it. Work not too 'ard, and' ll bring you in a dollar and a 'alf a day." But "it" was the matter in hand, and presently its nature became evident. At the corner of Fourteenth Street and Eighth Avenue Mr. Honeybun pointed across to a handsome white-stone building, whose very solidity inspired confidence. Tom could read for himself that it was a savings bank.
- 56. "Now what I'd do if it was my wad is this. I'd put three hundred and twenty-five of it in that there bank, which'd leave yer more'n twenty- five for yer eddication. But yer principal, no one won't be able to touch it but yerself, and twice a year yer'll be gettin' yer interest piled up on top of it." Tom's heart leaped. He had long meditated on savings banks. They had been part of his queer vision. To become "something big" he would have to begin by opening some such account as this. With Mr. Honeybun's proposal he felt as if he had suddenly grown taller by some inches, and older by some years. "You'll come over with me, won't you?" Mr. Honeybun demurred. "Well, yer see, kid, I'm a pretty remarkable character in this neighborhood. There's lots knows Honey Lem; and if they was to see me go in with you they might think as yer hadn't come by your dough quite hon—I mean, accordin' to yer conscience —or they might be bad enough to suppose as there was a put-up job between us. When I puts a few dollars into my own savings bank —I'm a savin' bird, I am—I goes right over to Brooklyn, where there ain't no wicked mind to suspeck me. So go in by yerself, and say yer wants to open a account. If anyone asks yer, tell him just how the money come to yer, and I don't believe as yer'll run no chanst of no one not believin' yer." So it was done. Tom came out of the building with his bank book buttoned into his breast pocket, and a conscious enhancement of life. "And now," Mr. Honeybun suggested, "we'll make tracks for Pappa's and eat." The "check," like the meal, was light, and Mr. Honeybun paid it. Tom protested, since he had money of his own, but his host took the situation gracefully. "Lord love yer, kid, ain't I yer next o' kin, as long as yer guv'nor's away? Who sh'd buy yer a lunch if it wasn't me?"
- 57. Childhood is naturally receptive. As Romulus and Remus took their food from a wolf when there was no one else to give it them, so Tom Quidmore found it not amazing to be nourished, first by a murderer, and then by a thief. It became amazing, a few years later, on looking back on it; but for the moment murderer and thief were not the terms in which he thought of those who had been kind to him. Not that he didn't try. He tried that very afternoon. When his next o' kin had gone back to his job of lifting and heaving in the Gansevoort Market, he returned to the empty room. It was his first return to it alone. When he had gone up from his breakfast in the chop saloon both Goodsir and Honeybun had accompanied him. Now the emptiness was awesome, and a little sinister. He had slept there the previous night, slept fitfully that is, waking every half hour to listen for the shuffling footstep. He heard other footsteps, dragging, thumping, staggering, but they always passed on to the story above, whence would come a few minutes later the sound of heavy boots thrown on the floor. Now and then there were curses, or male voices raised in a wrangle, or a few bars of a drunken song. During the earlier nights he had slept through these signals of Pappa's hospitality, or if he had waked, he knew that a grown-up man lay in the other bed, so that he was safe. Now he could only lie and shudder, till the sounds died down, and silence implied safety. He did his best to keep awake, so as to unlock the door the instant he heard a knock; but in spite of his efforts he slept. This return after luncheon brought him for the first time face to face with his state as a reality. There was no one there. It was no use going back to Bere, because there would be no one there. Rather than become again a State ward with the Tollivants, he would sell himself to slavery. What was he to do? The first thing his eye fell upon was his father's suitcase, lying open on the floor beside the bed, its contents in disorder. It was the way Quidmore kept it, fishing out a shirt or a collar as he needed one. The futility of this clothing was what struck the boy now. The
- 58. peculiar grief of handling the things intimately used by those who will never use them again was new to him. He had never supposed that so much sorrow could be stored in a soiled handkerchief. Stooping over the suitcase, he had accidentally picked one up, and burst into sudden tears. They were the first he had actually shed since he used to creep away to cry by himself in the heart-lonely life among the Tollivants. It occurred to him now that he had not cried when his adopted mother disappeared. He had not especially mourned for her. While she had been there, and he was daily face to face with her, he had loved her in the way in which he loved so easily when anyone opened the heart to him; but she had been no part of his inner life. She was the cloud and sunshine of a day, to be forgotten in the cloud and a sunshine of the morrow. Of the two, he grieved more for the man; and the man was a murderer, and probably a suicide. Sitting on the edge of his bed, he used these words in the attempt to work up a fortifying moral indignation. It was then, too, that he called Mr. Honeybun a thief. He must react against these criminal associations. He must stand on his own feet. He was not afraid of earning his own living. He had heard of boys who had done it at an age even earlier than thirteen, and had ended by being millionaires. They had always, however, so far as he knew, had some sort of ties to connect them with the body politic. They had had the support of families, sympathies, and backgrounds. They hadn't been adrift, like that haunting ship which never knew a port, and none but the God of the Sea to keep her from foundering. He could have believed in this God of the Sea. He wished there had been such a God. But the God that was, the God who was shut up in churches and used only on Sundays, was not of much help to him. Any help he got he must find for himself; and the first thing he must do would be to break away from these low-down companionships. And just as, after two or three hours of meditation, he had reached this conclusion, a tap at the door made him start. Quidmore had
- 59. come back! But before he could spring to the door it was gently pushed open, and he saw the patch over the left eye. "Got away early, son. Now, seems to me, we ought to be out after them overalls." The boy stood blank. "What overalls?" "Why, for yer job to-morrow. Yer can't work in them good clo'es. Yer'd sile 'em." In a second-hand shop, known to Honey Lem, in Charles Street, they found a suit of boy's overalls not too much the worse for wear. Honey Lem pulled out a roll of bills and paid for them. "But I've got my own money, Mr. Honeybun." "Dooty o' next o' kin, boy. I ain't doin' it for me own pleasure. Yer'll need yer money for yer eddication. Yer mustn't forgit that." The overalls bound him more closely to the criminal from whom he was trying to cut loose. More closely still he found himself tied by the scraps of talk he overheard between the former pals that evening. They were on the lowest of the steps leading up from the chop saloon, where all three of them had dined. Tom, who had preceded them, stood on the sidewalk overhead, out of sight and yet within earshot. "I tell yer I can't, Goody," Mr. Honeybun was saying, "not as long as I'm next o' kin to this 'ere kid. 'Twouldn't be fair to a young boy for me to keep no such company." Mr. Goodsir made some observation the nature of which Tom could only infer from Mr. Honeybun's response. "Well, don't yer suppose it's a damn sight 'arder for me to be out'n a good thing than it is for you to see me out'n it? I don't go in for no renounciation. But when yer've got a fatherless kid on yer 'ands ye' must cut out a lot o' nice stuff that'll go all right when yer've only yerself to think about. Ain't yer a Christian, Goody?" Once more Mr. Goodsir's response was to Tom a matter of surmise.
- 60. "Well, then, Goody, if yer don't like it yer can go to E and double L. What's more, I ain't a-goin' to sleep in our own room to-night, nor any night till that guy comes back. I'm goin' to sleep in the kid's room, and keep him company. 'Tain't right to leave a young boy all by hisself in a 'ouse like this, as full o' toughs as a ward'll be full o' politicians." Tom removed himself to a discreet distance, but the knowledge that the other bed in his room would not remain so creepily vacant was consciously a relief. He slept dreamlessly that night, because of his feeling of security. In the morning, not long after four, he was wakened by a hand that rocked him gently to and fro. "Come, little shaver! Time to git up! Got to be on yer job at five." The job was in a market that was not exactly a market since it supplied only the hotels. Together with the Gansevoort and West Washington Markets, it seemed to make a focal point for much of the food on the continent of America. Railways and steamers brought it from ranches and farms, from plantations and orchards, from rivers and seas, from slaughter-stockades and cold-storage warehouses, from the north and the south and the west, from the tropics and farther than the tropics, to feed the vast digestive machine which is the basis of New York's energies. Tom's job was not hard, but it was incessant. His was the duty of collecting and arranging the empty cases, crates, baskets, and coops, which were dumped on the raised platform surrounding the building on the outside, or which cluttered the stalls within. Trucks and vans took them away full on one day, and brought them back empty on another. It was all a boy could do to keep them stacked, and in order, according to sizes and shapes. The sizes in the main were small; the shapes were squares and oblongs and diminishing churnlike cylinders. Nimbleness, neatness, and goodwill were the requisites of the task, and all three of them the boy supplied. Fatigue that night made him wakeful. His companion in the other bed was wakeful too. In talking from bed to bed Tom found it a comfort to be dealing with an easy conscience. Mr. Honeybun had
- 61. nothing on his mind, nor was he subject to nightmares. Speculation on the subject of Quidmore's disappearance, and possible fate, turned round and round on itself, to begin again with the selfsame guesses. "And there's another thing," came from Mr. Honeybun. "If he don't come back, why, you'll come in for a good bit o' proputty, won't yer? Didn't he own that market-garden place, out there on the edge of Connecticut?" "He left it to his sister. He told me that the other night. You see, I wasn't his real son. I wasn't his son at all till about a year ago." This statement coming to Mr. Honeybun as something of a shock, Tom was obliged to tell the story of his life to the extent that he knew it. The only details that he touched on lightly were those which bore on the manner in which he had lost his "mudda." Even now it was difficult to name her in any other way, because in no other way had he ever named her. Obliged to blur the outlines of his earliest recollections, which in themselves were clear enough, his tale was brief. "So yer real name is Whitelaw," Mr. Honeybun commented, with interest. "I never hear that name but once. That was the Whitelaw baby. Ye'll have heard tell o' that?" Since Tom had never heard tell of the Whitelaw baby, the lack in his education was supplied. The Whitelaw baby had been taken out to the Park on a morning in May, and had vanished from its carriage. In the place where it had lain was found a waxen image so true in likeness to the child himself that only when it came time to feed him did the nurse make the discovery that she had wheeled home a replica. The mystery had been the source of nation-wide excitement for the best part of two years. It was talked of even now. It couldn't have been more than three or four years earlier that Mr. Honeybun had seen a daily paper, bearing the headlines that Harry Whitelaw had been found, selling like hotcakes to the women shopping in Twenty-third Street.
- 62. "And was he?" Tom asked, beginning at last to be sleepy. "No more'n a puff of tobacker smoke when yer'd blowed it in the air. The father, a rich banker—a young chap he was, too, I believe—he offers a reward of fifty thousand dollars to anyone as'd put him on the track o' the gang what had kidnapped the young 'un; and every son of a gun what thought he was a socialist was out to win the money. This 'ere Goody, he had a scheme. Tried to work me in on it, and I don't know but what I might a took a 'and if a chum o' mine hadn't got five year for throwin' the same 'ook without no bait on it. They 'auled in another chap I knowed, what they was sure he had somethink to do with it, and tried to make him squeal; but—" A long breath from Tom interrupted this flow of narrative. "Say, kiddy, yer ain't asleep, are yer? and me tellin' yer about the Whitelaw baby?" "I am nearly," the boy yawned. "Good night—Honey! Wake me in time in the morning." "That's a good name for yer to call me," the next o' kin commended. "I'll always be Honey to you, and you'll be Kiddy to me; and so we'll be pals. Buddies they call it over here." Echoes of a street brawl reached them through the window. Had he been alone, the country lad of thirteen would have shivered, even though the night was hot. But the knowledge of this brawny companion, lying but a few feet away, nerved him to curl up like a puppy, and fall asleep trustfully. XXI THE next two or three nights were occasions for the interchange of confidence. During the days the new pals saw little of each other, and sometimes nothing at all. With the late afternoon they could "clean themselves," and take a little relaxation. For this there was no great range of opportunity. Relaxation for Lemuel Honeybun had
- 63. hitherto run in directions from which he now felt himself cut off. He knew of no others, while the boy knew of none of any kind. "I tell yer, Goody," Tom overheard, through the open door of the room back of Pappa's, one day while he was climbing the stairs, "I ain't a-goin' to go while I've got this job on me hands. The Lord knows I didn't seek it. It's just one of them things that's give yer as a dooty, and I'm goin' to put it through. When Quidmore's come back, and it's all over, I'll be right on the job with the old gang again; but till he does it's nix. Yer can't mean to think that I don't miss the old bunch. Why, I'd give me other eye...." Tom heard no more; but the tone of regret worried him. True, if he wanted to break the bond this might be his chance. On the other hand, the thought of being again without a friend appalled him. While waiting in the hope that Quidmore might come back, the present arrangement was at least a cosy one. Nevertheless, he felt it due to his spirit of independence to show that he could stand alone. He waited till they were again lying feet to feet by the wall, and the air through the open window was cool enough to allow of their being comfortable, before he felt able to take an offhand, man-to-man tone. "You know, Honey, if you want to beat it back to your old crowd, I can get along all right. Don't hang round here on my account." "Lord love you, Kiddy, I know how to sackerfice meself. If I'm to be yer next o' kin, I'll be it and be damned. Done 'arder things than this in me life, and pulled 'em off, too. I'll stick to yer, kid, as long as yer wants me, if I never have another nice time in my life, and never see another quart bottle." The pathos of the life for which he might be letting himself in turned his thoughts backward over his career. "Why, if I'd 'a stuck at not puttin' others before meself I might still 'a been a gasfitter in Liverpool, Eng. That's where I was born. True 'eart-of-oak Englishman I was. Some people thinks they can tell it in the way I talk. Been over 'ere so long, though, seems to me I 'andle
- 64. the Yankee end of it pretty good. Englishman I met the other day— steward on one of the Cunarders he was—said he wouldn't 'a knowed me from a born New Yorker. Always had a gift for langwidges. Used to know a Frenchman onst; and I'll be 'anged if I wasn't soon parley-vooin' with him till he'd thought I was his mother's son. But it's doin' my dooty by others as has brought me where I am, and I don't make no complaint of it. Job over at the Gansevoort whenever I wants one, which ain't always. Quite a tidy little sum in the savings bank in Brooklyn. Friends as'll stick by me as long as I'll stick by them. And if I hadn't lost me eye—but how was I to know that that low-down butler was a-layin' for me at the silver- pantry door, and' d let me have it anywhere he could 'it me?... And when that eyeball cracked, why, I yelled fit to bring the whole p'lice- force in New York right atop o' me." Tom was astounded. "But you said you lost your eye saving a young lady's life." Mr. Honeybun's embarrassment lasted no more than the time needed for finding the right words. "Oh, did I? Well, that was the other side of it. Yer've heard that there's always two sides to a story, haven't yer? I can't tell yer both sides to onst, now can I?" He judged it best, however, to revert to the autobiographical. The son of a dock hand in Liverpool, he had been apprenticed to a gasfitter at the age of seventeen. "But my genius was for somethink bigger. I didn't know just what it'd be, but I could see it ahead o' me, all wuzzy-like. After a bit I come to know it was to fight agin the lor o' proputty. Used to seem to me orful to look around and see that everythink was owned by somebody. Took to goin' to meetin's, I did. Found out that me and me class was the uninherited. 'Gord,' I says to meself then, 'I'll inherit somethink, or I'll bust all Liverpool.' Well, I did inherit somethink—inherited a good warm coat what a guy had left to mark his seat in the Midland Station. Got away with it, too. Knowin' it was
- 65. mine as much as his, I walks up and throws it over my arm. Ten minutes later I was a-wearin' of it in Lime Street. That was the beginnin', and havin' started in, I begun to inherit quite a lot o' things. 'Nothink's easier,' says I, 'onst you realizes that the soul o' man is free, and that nothink don't belong to nobody.' Fightin' for me class, I was. Tried to make 'em see as they ought to stop bein' the uninherited, and get a move on—and the first thing I know I was landed in Walton jail. You're not asleep, Kiddy, are you?" Not being asleep, Tom came in for the rest of the narrative. Released from Walton jail, Mr. Honeybun had "made tracks" for America. "Wanted to git away from a country where everythink was owned, and find the land o' the free. But free! Lord love yer, I hadn't been landed a hour before I see everythink owned over 'ere as much as it is in a back'ard country like old England. Let me tell you this, Kid. Any man that thinks that by comin' to America he'll git somethink for nothink'll find hisself sold. I ain't had nothink except what I've worked for—or collared. Same old lor o' proputty what's always been a injustice to the pore. Had to begin all over agin the same old game of fightin' it. But what's a few months in chokey when you're doin' it for yer feller creeters, to show 'em what their rights is?" A few nights later Tom was startled by a new point of view as to his position. "I've been thinkin', Kiddy, that since yer used to be a State ward, yer'll have to be a State ward agin, if the State knows you're knockin' round loose." The boy cried out in alarm. "Oh, but I won't be. I'll kill myself first." He could not understand this antipathy, this horror. In a mechanical way the State had been good to him. The Tollivants had been good to him, too, in the sense that they had not been unkind. But he could not return to the status. It was the status that dismayed him. In Harfrey it had made him the single low-caste individual in a prim and high-caste world, giving everyone the right to disdain him. They couldn't help disdaining him. They knew as well as he did that in
- 66. principle he was a boy like any other; but by all the customs of their life he was a little pariah. Herding with thieves and murderers, it was still possible to respect himself; but to go back and hang on to the outer fringe of the organized life of a Christian society would have ravaged him within. He said so to Honeybun energetically. "That's the way I figured that yer'd feel. So long as you're on'y waitin'—or yer can say that you're on'y waitin'—till yer pop comes back, it won't matter much. It'll be when school begins that it'll go agin yer. There's sure to be some pious woman sneepin' round that'll tell someone as you're not in school when you're o' school age, and then, me lad, yer'll be back as a State ward on some down-homer's farm." Tom lashed the bed in the darkness. "I won't go! I won't go!" "That's what I used to say the first few times they pinched me; but yer'll jolly well have to go if they send yer. Now what I was thinkin' is this. It's in New York State that yer'd be a State ward. If you was out o' this State there'd be all kinds o' laws that couldn't git yer back again. Onst when I'd been doin' a bit o' socializin' in New Jersey, and slipped back to Manhattan—well, you wouldn't believe the fuss it took to git me across the river when the p'lice got wind it was me. Never got me back at all! Thing died out before they was able to fix up all the coulds and couldn'ts of the lor." He allowed the boy to think this over before going on with his suggestion. "Now if you and me was to light out together to another State, they wouldn't notice that we'd gone before we was safe beyond their clutches. If we was to go to Boston, say! Boston's a good town. I worked Boston onst, me and a chap named...." The boy felt called on to speak. "I wouldn't be a socialist, not if it gave me all Boston for my own." The statement, coming as it did, had the vigor of an ultimatum. Though but a repetition of what he had said a few days before, it was a repetition with more force. It was also with more significance,
- 67. fundamentally laying down a condition which need not be discussed again. After long silence Mr. Honeybun spoke somewhat wistfully. "Well, I dunno as I'd count that agin yer. I sometimes thinks as I'll quit bein' a socialist meself. Seems to me as if I'd like to git back with the old gang, and be what they calls a orthodock. You know what a orthodock is, don't yer?" "It's a kind of religion, isn't it?" "It ain't so much a kind of religion as it's a kind o' way o' thinkin'. You're a orthodock when you don't think at all. Them what ain't got no mind of their own, what just believes and talks and votes and lives the way they're told to, they're the orthodocks. It don't matter whether it's religion or politics or lor or livin', the people who don't know nothink but just obeys other people what don't know nothink, is the kind that gits into the least trouble." "Yes, but what do you want to be like that for? You have got a mind of your own." "Well, there's a good deal to be said, Kiddy. First there's you." "Oh, if it's only me...." "Yes, but when I'm yer next o' kin it isn't on'y you; it's you first and last. I got to bring you up an orthodock, if I'm going to bring you up at all. Yer can't think for yerself yet. You're too young. Stands to reason. Why, I was twenty, and very near a trained gasfitter, before I'd begun thinkin' on me own. What yer does when yer're growed up'll be no concern o' mine. But till you are growed up...." Tom had heard of quicksands, and often dreamed that he was being engulfed in one. He had the sensation now. Circumstances having pushed him where he would not have ventured of his own accord, the treacherous ground was swallowing him up. He couldn't help liking Honey Lem, since he liked everyone in the world who was good to him; he was glad of his society in these lonely nights, and of the sense of his comradeship in the background even in the day; but
- 68. between this gratitude and a lifelong partnership he found a difference. There were so many reasons why he didn't want permanent association with this fairy godfather, and so many others why he couldn't find the heart to tell him so! He was casting about for a method of escape when the fairy godfather continued. "This 'ere socialism is ahead of its time. People don't understand it. It don't do to be ahead o' yer time, not too far ahead, it don't. Now I figure out that if I was to go back a bit, and git in among them orthodocks, I might do 'em good like. Could explain to 'em. I ain't sure but what I've took the wrong way, showin' 'em first, and explainin' to 'em afterwards. Now if I was to stop showin' 'em at all, and just explain to 'em, why, there'd be folks what when I told 'em that nothink don't belong to nobody they'd git the 'ang of it. Begins to seem to me as if I'd done me bit o' sufferin' for the cause. Seen the inside o' pretty near every old jug round New York. It's aged me. But if I was to sackerfice me opinions, and make them orthodocks feel as I was one of 'em, I might give 'em a pull along like." The next day being Sunday, they slept late into the morning. In the afternoon Honey Lem had a new idea. Without saying what it was, he took the boy to walk through Fourteenth Street, till they reached Fifth Avenue. Here they climbed to the top of an electric bus going northward, and Tom had a new experience. Except for having crossed it in the market lorry, in the dimness and emptiness of dawn, this stimulating thoroughfare was unknown to him. Even on a Sunday afternoon in summer, when shops were shut, residences closed, and saunterers relatively few, it added a new concept to those already in his mental possession. It was that of magnificence. These ornate buildings, these flashing windows, these pictures, jewels, flowers, fabrics, furnishings, did more than appeal to his eye. They set free a function of his being that had hitherto been sealed. The first atavistic memory of which he had ever been aware was consciously in his mind. Somewhere, perhaps in some life before he was born, rich and beautiful things had been his accessories. He had been used to them. They were not a surprise to
- 69. him now; they came as a matter of course. To see them was not so much a discovery as it was a return to what he had been accustomed to. He was thinking of this, with an inward grin of derision at himself for feeling so, when Honey went back to the topic of the night before. "The reason I said Boston is because they've got that great big college there. If I'm to bring yer up, I'll have to send yer to college." The opening was obvious. "But, Honey, you don't have to bring me up." "How can I be yer next o' kin if I don't bring ye' up, a young boy like you? Be sensible, Kiddy. Yer ch'ice is between me and the State, and I'd be a lot better nor that, wouldn't I? The State won't be talkin' o' sendin' yer to college, mind that now." There was no controverting the fact. As a State ward, he would not go to college, and to college he meant to go. If he could not go by one means he must go by another. Since Honey would prove a means of some sort, he might be obliged to depend on him. The bus was bowling and lurching up the slope by which Fifth Avenue borders the Park, when Honey rose, clinging to the backs of the neighboring seats. "We'll git out at the next corner." Having reached the ground, he led the way across the street, scanning the houses opposite. "There it is," he said, with choked excitement, when he had found the façade he was looking for. "That big brown front, with the high steps, and the swell bow-winders. That's where the Whitelaw baby used to live." Face to face with the spot, Tom felt a flickering of interest. He listened with attention while Honey explained how the baby carriage had for the last time been lifted down by two footmen, and how it was wheeled away by the nurse. "Nash, her name was. I seen her come out one day, when Goody and me was standin' 'ere. Nice little thing she seemed, English, same
- 70. as I be. Yes, Goody and me'd sniggle and snaggle ourselves every which way to see how we could cook up a yarn that'd ketch on to some o' that money. We sure did read the papers them days! There wasn't nothink about the Whitelaw baby what we didn't know. Now, if yer've looked long enough at the 'ouse, Kid, I'll show yer somethink else." They went into the Park by the same little opening through which the Whitelaw baby had passed, not to return. Like a detective reconstructing the action of a crime, he followed the path Miss Nash had taken, almost finding the marks of the wheels in the gravel. Going round the shoulder of a little hill, they came to a fan-shaped elm, in the shade of which there was a seat. Beyond the seat was a clump of lilac, so grouped as to have a hollow like a horseshoe in its heart, with a second seat close by. Honey revived the scene as if he had witnessed it. Miss Nash had sat here; her baby carriage had stood there. The other nurse, name o' Miss Messenger, had put her baby beneath the elm, and taken her seat where she could watch it. All he was obliged to leave out was the actual exchange of the image for the baby, which remained a mystery. "This 'ere laylock bush ain't the same what was growin' 'ere then. That one was picked down, branch by branch, and carried off for tokens. Had a sprig of it meself at one time. I always thinks them little memoriums is instructive. I recolleck there was a man 'anged in Liverpool, and the 'angman, a friend of my guv'nor's, give me a bit of the chap's shirt, what he'd left in his cell when he changed to a clean one to be 'anged in. Well, I kep' that bit o' shirt for years. Always reminded me not to murder no one. Wish I had it now. Funny it'd be, wouldn't it, if you turned out to be the Whitelaw baby? He'd a' been just about your age." Tom threw himself sprawling on the seat where Miss Nash had read Juliet Allingham's Sin, and laughed lazily. "I couldn't be, because his name was Harry, and mine's Tom." "Oh, a little thing like that wouldn't invidiate your claim."
- 71. "But I haven't got a claim. You don't suppose my mother stole me, do you? That's the very thing she used to tell me not to...." The laugh died on his lips. As Honey stood looking down at him there was a light in his blue-gray eye like the striking of a match. Tom knew that the same thought was in both their minds. Why should a woman have uttered such a warning if she had not been afraid of a suspicion? A flush that not only reddened his tanned cheeks, but mounted to the roots of his bushy, horizontal eyebrows, made him angry with himself. He sprang to his feet. "Look here, Honey! Aren't there animals in this Park? Let's go and find them." To his relief, Honey pressed no question as to his mother and stolen babies as they went off to the Zoo. XXII THE move to Boston was made during August, so that they might be settled in time for the opening of the schools. The flitting was with the ease of the obscure. Also with the ease of the obscure, Lemuel changed his name to George, while Tom Quidmore became again Tom Whitelaw. There were reasons to justify these decisions on the part of both. "Got into trouble onst in Boston under the name of Lemuel, and if any old sneeper was to look me up.... Not but what Lemuel isn't a more aristocraticker name than George; but there's times when somethink what no one won't notice'll suit you best. So I'll be George Honeybun, a pal o' yer father's, what left yer to me on his dyin' deathbed." The name of Tom Whitelaw was resumed on grounds both sentimental and prudential. In the absence of any other tie to the human race, it was something to the boy to know that he had had a
- 72. father. His father had been a Whitelaw; his grandfather had been a Whitelaw; there was a whole line of Whitelaws back into the times when families first began to be known by names. A slim link with a past, at least it was a link. The Quidmore name was no link at all; it was disconnection and oblivion. It signified the ship that had never had a port. As a Whitelaw, he had sailed from somewhere, even though the port would forever be unknown to him. It was a matter of prudence, too, to cover up his traces. In the unlikely event of the State of New York busying itself with the fate of its former ward, the name of Quidmore would probably be used. A well-behaved Tom Whitelaw, living with his next of kin, and attending school in Boston according to the law, would have the best chance of going unmolested. They found a lodging, cheap, humble, but sufficient, on that northern slope of Beacon Hill which within living memory has more than once changed hands with the silent advance and recession of a tide coming in and going out. There are still old people who can remember when some of the worthiest of the sons of the Puritans had their windows, in these steep and narrow streets, brightened by the rising or the setting sun. Then, with an almost ghostly furtiveness, they retired as the negro came and routed them. The negro seemed fixed in possession when the Hebrew stole on silently, and routed him. At the time when George Honeybun and Tom Whitelaw came looking for a home, the ancient inhabitant of the land was beginning to creep back again, and the Hebrew taking flight. In a red-brick house of forbidding expression in Grove Street they found a room with two beds. Within a few days Honey, whose strength was his skill, was working as a stevedore on the Charlestown docks. Tom was picking up small jobs about the markets. By September he had passed his examinations and had entered the Latin School. A new life had begun. From the old life no pursuit or interference ever followed them.
- 73. The boy shot up. In the course of a year he had grown out of most of his clothes. To the best of his modest ability, Honey was generous with new ones. He was generous with everything. That Tom should lack nothing, he cut down his own needs till he seemed to have none but the most elemental. Of his "nice times" in New York nothing had followed him to Boston but a love of spirits and tobacco. Of the two, the spirits went completely. When Tom's needs were pressing the supply of tobacco diminished till it sometimes disappeared. If on Sundays he could venture over the hill, to listen to the band on the Common, or stroll with the boy in the Public Gardens, it was because the Sunday suit, bought in the days when he had no one to provide for but himself, was sponged and pressed and brushed and mended, with scrupulous devotion. The motive of so much self-denial puzzled Tom, since, so far as he could judge, it was not affection. He was old enough now to perceive that affection had inspired most of his good fortune. People were disposed to like him for himself. There was rarely a teacher who did not approve of him. By the market men, among whom he still picked up a few dollars on Saturdays and in vacations, he was always welcomed heartily. In school he never failed to hold his own till the boys discovered that his father, or uncle, or something, was a stevedore, after which he was ignored. Girls regarded him with a hostile interest, while toward them he had no sentiments of any kind. He could go through a street and scarcely notice that there was a girl in it, and yet girls wouldn't leave him alone. They bothered him with overtures of friendship to which he did not respond, or tossed their heads at him, or called him names. But in general the principle was established that he could be liked. But Honey was an enigma. Love was apparently not the driving power urging him to these unexpected fulfillments. If it was, it had none of the harmless dog-and-puppy ways which Tom had grown accustomed to. Honey never pawed him, as the masters often pawed the boys, and the boys pawed one another. He never threw an arm across his shoulder, or called him by a more endearing name
- 74. than Kiddy. Apart from an eagle-eyed solicitude, he never manifested tenderness, nor asked for it. That Tom would ever owe him anything he didn't so much as hint at. "Dooty o' next o' kin" was the blanket explanation with which he covered everything. "But you're not my next of kin," Tom, to whom schooling had revealed the meaning of the term, was bold enough to object. "Next of kin means that you'd be my nearest blood relation; and we're not relations at all." Honey was undisturbed in his Olympian detachment. "Do yer suppose I dunno that? But I believes as Gord sees we're kin lots o' times when men don't take no notice. You was give to me. You was put into my 'ands to bring up. And up I'm goin' to bring yer, if it breaks me." It was a close Sunday evening in September, the last of the summer holidays. Tom would celebrate next day by entering on a higher grade at school. He had had new boots and clothes. For the first time he was worried by the source of this beneficence. As night closed down they sat for a breath of fresh air on the steps of the house in Grove Street. Grove Street held the reeking smell of cooking, garbage, and children, which only a strong wind ever blows away from the crowded quarters of the cities, and there had been no strong wind for a week. Used to that, they didn't mind it. They didn't mind the screeching chatter or the raucous laughter that rose from doorways all up and down the hill, nor the yelling of the youngsters playing in the roadway. Somewhere round a corner a group of Salvationists, supported by a blurting cornet, sang with much gusto: Oh, how I love Jesus! Oh, how I love Jesus! Oh, how I love Jesus! Because He first loved me. They didn't mind it when Mrs. Danker, their landlady, a wiry New England woman, sitting in the dark of the hall behind them, joined in, in her cracked voice, with the Salvationists, nor when Mrs.
- 75. Gribbens, a stout old party who picked up a living scrubbing railway cars, joined in with Mrs. Danker. From neighboring steps mothers called out to their children in Yiddish, and the children answered in strident American. But to Honey and Tom all this was the friendly give-and-take of promiscuity which they would have missed had it not been there. Each was so concentrated on his own ruling purpose that nothing external was of moment. Honey was to give, and Tom was to receive, an education. That the recipient's heart should be fixed on it, Tom found natural enough; but that the giver's should be equally intense seemed to have nothing to account for it. He glanced at the quiet figure, upright and muscular, his hands on his knees, like a stone Pharaoh on the Nile. "Why don't you smoke?" "I don't want to drop no ashes on this 'ere suit." "Have you got any tobacco?" "I didn't think to lay in none when I come 'ome yesterday." "Is that because there was so much to be spent on me?" "Oh, I dunno about that." Tom gathered all his ambitions together and offered them up. "Well, I guess this can be the last year. After I've got through it I'll be ready to go to work." "And not go to college!" The tone was one of consternation. "Lord love yer, Kiddy, what's bitin' yer now?" "It's biting me that you've got to work so hard." "If it don't bite me none, why not let it go at that?" "Because I don't seem able to. I've taken so much from you." "Well, I've had it to 'and out, ain't I?" "But I don't see why you do it."
- 76. "A young boy like you don't have to see. There's lots o' things I didn't understand at your age." "You don't seem specially—" he sought for words less direct, but without finding them—"you don't seem—specially fond of me." "I never was one to be fond o' people, except it was a dog. Always had a 'ankerin' for a dog; but a free life don't let yer keep one. A dog'll never go back on yer." "Well, do you think I would?" "I don't think nothink about it, Kid. When the time comes that you can do without me...." "That time'll never come, Honey, after all you've done for me." "I don't want yer to feel yerself bound by that." "I don't feel myself bound by it; but—dash it all, Honey!—whatever you feel or don't feel about me, I'm fond of you." He was still imperturbable. "Well, Kid, you wouldn't be the first, not by a lot." "But if I can never be anything for you, or do anything for you...." "There's one thing you could do." "What is it? I don't care how hard it is." "Well, when you're one o' them big lawyers, or bankers, or somethink—drorin' yer fifty dollars a week—you can have a shy at this 'ere lor o' proputty. It don't seem right to me that some people should have all the beef to chaw, and others not so much as the bones; but I can't git the 'ang of it. If nothink don't belong to nobody, then what about all your dough in the New York savin's bank, and mine in the one in Brooklyn? We're keepin' it agin yer goin' to college, ain't we? And don't that belong to us? Yes, by George, it do! So there you are. But if when yer gits yer larnin' yer can steddy it out...."
- 77. XXIII THE boy was adolescent, sentimental, and lonely. Mere human companionship, such as that which Honey gave him, was no longer enough for him. He was seeing visions and dreaming dreams. He began to wish he had some one with whom to share his unformulated hopes, his crude and burning opinions. He looked at fellows who were friends going two and two, pouring out their foolish young hearts to each other, and envied them. The lads of his own age liked him well enough. Now and then one of them would approach him with shy or awkward signals, making for closer acquaintance; but when they learned that he lived in Grove Street with a stevedore they drew away. None of them ever transcended the law of caste, to stand by him in spite of his humble conditions. Boys whose families were down wanted nothing to hamper them in climbing up. Boys whose families were up wanted nothing that might loosen their position and pull them down. The sense of social insecurity which was the atmosphere of homes reacted on well- meaning striplings of fifteen, sixteen, and seventeen, turning them into snobs and cads before they had outgrown callowness. But during the winter of the year in which he became sixteen there were two, you might have said three, who broke in upon this solitude. In walking to the Latin School from Grove Street he was in the habit of going through Louisburg Square. If you know Boston you know Louisburg Square as that quaint red-brick rectangle, like many in the more Georgian parts of London, which commemorates the gallant dash of the New England colonists on the French fortress of Louisburg in Cape Breton. It is the heart of that conservative old Boston, which is now shrinking in size and importance before the onset of the foreigner till it has become like a small beleaguered citadel. Here the descendants of the Puritans barricade themselves behind their financial walls, as their ancestors within their stockades, while their city is handed over to the Irishman and the Italian as an
- 78. undefended town. The Boston of tradition is a Boston of tradition only. Like the survivors of Noah's deluge clinging to the top of a rock, they to whom the Boston of tradition was bequeathed are driven back on Beacon Hill as a final refuge from the billows rising round them. A high-bred, cultivated, sympathetic people, they have so given away their heritage as to be but a negligible factor in the State, in the country, of which their fathers and grandfathers may be said once to have kept the conscience. But to Tom Whitelaw Louisburg Square meant only the dignified fronts and portals behind which lived the rich people who had no point of contact with himself. They couldn't have ignored him more completely than he ignored them. He thought of them as little as the lion cub in a circus parade thinks of the people of the city through which he passes in processions. Then, one day, one of these strangers spoke to him. It was a youth of about his own age. More than once, as Tom went by, and the stout boy stood on the sidewalk in front of his own house, they had looked each other up and down with unabashed mutual appraisal. Tom saw a lad too short for his width, and unhealthily flabby. He had puffy hands, and puffy cheeks, with eyes seeming smaller than they were because the puffy eyelids covered them. The mouth had those appealing curves comically troubled in repose, but fulfilling their purpose in giggling. On the first occasion when Tom passed by the lips were set to the serious task of inspection. They said nothing; they betrayed nothing. Tom himself thought nothing, except that the boy was fat. They had looked at each other some two or three times a week, for perhaps a month, when one day the fat boy said, "Hullo!" Tom also said, "Hullo!" continuing on his way. A day or two later they repeated these salutations, though neither forsook his attitude of reserve. The fat boy did this first, speaking when they had hullo'ed each other for the third or fourth time. His voice was high and girlish, and yet with a male crack in it. "What school do you go to?"
- 79. Tom stopped. "I go to the Latin School. What school do you go to?" "I go to Doolittle and Pray's." "That's the big private school in Marlborough Street, isn't it?" The fat boy made the inarticulate grunt which with most Americans means "Yes." "I was put down for Groton, only mother wouldn't let me leave home. I'm going to Harvard." "I'm going to Harvard, too. What class do you expect to be in?" The fat boy replied that he expected to be in the class of nineteen- nineteen. Tom said he expected to be in that class himself. "Now I've got to beat it to the Latin School. So long!" "So long!" Tom carried to his school in the Fenway an unusual feeling of elation. With friendly intent someone had approached him from the world outside. It was not the first time it had ever happened, but it was the first time it had ever happened in just this way. He could see already that the fat boy was not one of those he would have chosen for a friend; but he was so lonely that he welcomed anyone. Moreover, he divined that the fat boy was lonely, too. Boys of that type, the Miss Nancy and the mother's darling type, were often consumed by loneliness, and no one ever pitied them. Few went to their aid when other boys "picked" on them, but of those few Tom Whitelaw was always one. He found them, once you had accepted their mannerisms, as well worth knowing as other boys, while they spared him a scrap of admiration. It was possible that in this fat boy he might find the long-sought fellow who would not "turn him down" on discovering that he lived in Grove Street. Being turned down in this way had made him sick at heart so often that he had decided never any more to make or trust advances. In suffering temptation again he assured himself that it would be for the last time in his life.
- 80. On returning from school he looked for the boy in Louisburg Square, but he was not there. A few hundred yards farther, however, he came in for another adventure. The January morning had been mild, with melting snow. By midday the wind had shifted to the north, with a falling thermometer. By late afternoon the streets were coated with a glaze of ice. Tom could swagger down the slope of Grove Street easily enough in the security of rubber soles. But not so a girl, whose slippers and high French heels made her helpless on the steep glare. Having ventured over the brow of the hill, she found herself held. A step into the air would have been as easy as another on this slippery descent. The best she could do was to sway in the keen wind, keeping her balance with the grace of one of the blue spruces which used to be blown about at Bere. Her outstretched arms waved up and down, as a blue spruce waves its branches. Coming abreast of her, Tom found her laughing to herself, but on seeing him she laughed frankly and aloud. "Oh, catch me! I'm going to tumble! Ow-w-w!" Tom snatched at one hand, while she caught him by the shoulder with the other. "Saved! Wasn't it lucky that you came along? You're the Whitelaw boy, aren't you?" Tom admitted that he was, though his new sensations, with this exquisite creature clinging to him like a drowning man to his rescuer, choked the monosyllable in his throat. Though he had often in a scrimmage protected little boys, he had never before been thrown into this comic, laughing tussle with a girl. It had the excuse for itself that she couldn't stand unless he held her up. He held her firmly, looking into her dancing eyes with his first emotional consciousness of a girl's prettiness. His arm supporting her, she ventured on a step. "I'm Maisie Danker," she explained, while taking it. "I see you going in and out the house."
- 81. "I've never seen you." "Perhaps you've seen me and not noticed me." "I couldn't," he declared, with vehemence. "I've never seen you before in my life. If I had...." Her high heels so nearly slipped from under her that they were compelled to hold each other as if in an embrace. "If you had— what?" He knew what, but the words in which to say it needed a higher mode of utterance. The red lips, the glowing cheeks, had the vitality of the lively eyes. A red tam-o'-shanter, a red knitted thing like a heavenly translation of his own earthly sweater, were bewitchingly diabolic when worn with a black skirt, black stockings, and black shoes. As he did not respond to her challenge, she went on with her self- introduction. "I guess you haven't seen me, because I only arrived three days ago. I'm Mrs. Danker's niece. Live in Nashua. Worked in the woolen mills there. Now I've come to visit my aunt for the winter." For the sake of hearing her speak, he asked if she was going to work in Boston. "I don't know. Maybe I'll take singing lessons. Got a swell voice." If again he was dumb it was because of the failure of his faculties. Nothing in his experience had prepared him for the give-and-take of a badinage in which the surface meanings were the less important. Foolish and helpless, unable to show his manly superiority except in the strength with which he held her up, he got a lesson in the new art there and then. "Ever dance?" "I'm never asked." "Oh, it's you that ought to do the asking."
- 82. "I mean that I'm never asked where there's dancing going on." "Gee, you don't have to be. You just find a girl—and go." "But I don't know how to dance." "I'll teach you." Slipping and sliding, with cries of alarm on her part, and stalwart assurances on his, they approached their own doorstep. "Ow-w-w! Hold me! I'm going!" "No you're not—not while I've got you." "But I don't want to grab you so hard." "That's all right. I can stand it." "But I can't. I'm not used to it." "Then it's a very good time to begin." "What's the use of beginning if there's nothing to go on with?" "How do you know there won't be?" "Well, what can there be?" Had Miss Danker always waited for answers to her questions Tom would have been more nonplussed than he was. But the game which he didn't know at all she knew thoroughly, according to her lights. She never left him at a loss for more than a few seconds at a time. Her method being that of touch-and-go, reserving to herself the right of coming back again, she carried his education one step farther still. "Don't you ever go to the movies?" He replied that he had gone once or twice with Honey, but not often. To be on the same breezy level as herself, he added in explanation: "Haven't got the dough." "But the movies don't take dough, not hardly any." "They take more than I've got."
- 83. "More than you've got? Gee! Then you can't have anything at all." It was not so much a taunt as it was a statement, and yet it was a statement with a little taunt in it. For once driven to bravado, he gave away a secret. "Well, I haven't—except what's in the bank." "Oh, you've got money in the bank, have you?" "Sure! But I'm keeping it to go to college." She stared at him as if he had been a duck-billed rabbit, or some variety of fauna hitherto unknown. "Gee! I should think a fellow who had money in the bank would want to blow some of it on having a good time—a fellow with any jazz." Once more she spared him discomfiture. Slipping into the hallway, she said over her shoulder as he followed her: "How old are you?" "Sixteen." She flashed round at him. "Sixteen! Gee! I thought you was my age if you was a day. Honest I did. I'm eighteen, an old lady compared with you." "Oh, but boys are always older than girls, for their age." "You are, sure. Anyways, you saved me on that slippery hill, and I think you ought to have a kiss for it. Come, baby, kiss your poor old ma." Though the hallway was dark, the kiss had to be given and taken furtively. Whatever it was to Maisie Danker, to Tom Whitelaw it was the entrance to a higher and an increased life. The pressure of her lips on his sent through his frame a dynamic glow he had not supposed to be among nature's possibilities. Moreover, it threw light on that experience as to which he had mused ever since he had first talked confidentially to Bertie Tollivant. Though instinct had taught him something in the intervening years, he had up to this minute
- 84. Welcome to Our Bookstore - The Ultimate Destination for Book Lovers Are you passionate about books and eager to explore new worlds of knowledge? At our website, we offer a vast collection of books that cater to every interest and age group. From classic literature to specialized publications, self-help books, and children’s stories, we have it all! Each book is a gateway to new adventures, helping you expand your knowledge and nourish your soul Experience Convenient and Enjoyable Book Shopping Our website is more than just an online bookstore—it’s a bridge connecting readers to the timeless values of culture and wisdom. With a sleek and user-friendly interface and a smart search system, you can find your favorite books quickly and easily. Enjoy special promotions, fast home delivery, and a seamless shopping experience that saves you time and enhances your love for reading. Let us accompany you on the journey of exploring knowledge and personal growth! ebookgate.com