![1 Introduction
It would not be too much of an exaggeration to say that oscillations are one of the
main forms of motion. They range from the periodic motion of planets to random
openings of ion channels in cell membranes. They are observed at various levels of
organization, have various origins and various properties. Since Newton’s crack at
the three-body problem and until just a few decades ago, the range of phenomena
regarded as oscillations were limited to damped, periodic and quasiperiodic oscil-
lations at best. A significant achievement of the second half of the 20th century is
the admission of deterministic chaos and noise-induced rhythms as equals into the
oscillation family.
Nature is not based on isolated individual systems. It is rich in connections, in-
teractions and communications of different kinds that are complex beyond belief.
With this, synchronization is the most fundamental phenomenon associated with
oscillations. It is a direct and widely spread consequence of the interaction of dif-
ferent systems with each other. In most general terms, synchronization means that
different systems adjust the time scales of their oscillations due to interaction, but
there is a large variety of its manifestations and of the accompanying fascinating
phenomena.
Anyone writing a book on synchronization is faced with two problems: on one
hand, one has to deal with a huge amount of material on the particular aspects and
effects; and on the other hand, there is a need to formulate a universal approach that
would embrace all the particular cases. Fortunately, an essential contribution to the
second problem has been made by Pikovsky, Rosenblum, and Kurths in their recent
book [214], that has provided a contemporary view on synchronization as a univer-
sal phenomenon that manifests itself in the entrainment of rhythms of interacting
self-sustained systems. This viewpoint is in agreement with the approach developed
since the time of Huygens, and is completely shared by ourselves. In writing the
present book we were motivated by the following considerations:
• Recently, a large variety of new synchronization phenomena were discovered
that are inherent in complex (chaotic) systems, but do not occur in simple peri-
odic oscillators. With the modern fascination for the beauty and the complexity
of the new effects, there is a tendency to forget about the basic phenomena and
theoretical results associated with “simply” periodic oscillations. This is largely
due to the fact that not all involved in the studies of these phenomena, and espe-
cially younger researchers and students, have the respective education. It turns](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-19-320.jpg)
![2 General Remarks
2.1 What Are We Going to Talk About?
“Synchronization, of course, but what is it and why should I bother?” you might
ask.
Look, everything around us is moving. As René Descartes used to say [71],
“Give me the matter and motion and I will construct the universe.” Others say “mo-
tion is the mode of existence of matter” [274]. How exactly is the matter moving?
One very popular possibility is the motion that demonstrates a certain degree of rep-
etition, this would be an oscillation. Your heart is an oscillator, can you hear how it
beats?
Now consider several oscillators and let them feel each other’s motion, no matter
how exactly—the scientists would say “couple them.” Most likely, the coupling will
not go unnoticed by any of these systems: all of them will change their behavior to
this or that extent. In fact, this is going on in your body right now: you inhale and
exhale repetitively, and thus influence the way your heart beats without knowing it
perhaps. The basic features of oscillations are their amplitude and shape, but when
we talk about repetition of anything, a natural question is “how often?” With this
question arises the concept of a characteristic time scale of oscillations.
What does coupling have to do with all this? Well, because of coupling all
aspects of the system’s behavior would generally change, sometimes most dras-
tically. So, before you couple anything that oscillates, it would be good to know
the possible consequences in advance, wouldn’t it? We can tell you right now that
a lot of things can happen. For example, oscillations can stop altogether, which
might be good sometimes, but occasionally disastrous. Or they could become to-
tally unpredictable—but you might like it nevertheless because it looks beautiful.
But the phenomenon which is most often associated with synchronization is the
change of the time scales of interacting systems: if you couple the systems cleverly,
they can start to oscillate “syn-chronously,” which means “sharing the same time”
[214]. For your heart and breathing this can mean that, say, while you are breathing
once, your heart makes three beats exactly.
One can say that synchronization is the most fundamental phenomenon that oc-
curs in oscillating processes. In most general terms, synchronization can be defined](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-24-320.jpg)
![12 2 General Remarks
2.3 Self-Sustained Oscillations: A Key Concept
in Synchronization Theory
Before we start talking about any synchronization at all, we need to outline more
precisely the class of systems and processes in which we can expect it to occur.
Systems that oscillate in principle are usually called oscillators. But the systems
we are interested in should be capable of demonstrating oscillations that are self-
sustained, or self-oscillations. The concept of self-oscillations was first proposed by
Andronov, Khaikin and Vitt1 in 1937 [12] (for the English version see [14]2). Self-
oscillations form a special, but rather broad class of all oscillating processes and are
characterized by the following features.
2.3.1 Features of Self-Oscillations
Below we list the features of self-oscillations.
• First and foremost, they do not damp, i.e., the repetitive motion of the system
does not stop with the course of time, and does not show the tendency to stop.3
• Second and equally important, they oscillate “by themselves,” i.e., not because
they are repetitively kicked from outside.
• The third feature is perhaps the most intriguing and fascinating: the shape, am-
plitude and time scale of these oscillations are chosen by the oscillating system
alone.4 An outsider cannot easily change them, e.g., by setting different initial
conditions.5
Examples of self-oscillators are a grandfather pendulum clock, a whistle, your throat
when you sing a musical note, as well as many musical instruments, your heart and
many other biological systems, a bottle of water with a narrow neck that is put
vertically with its neck down (water will come out in pulses). In order to prevent
possible confusion, we would like to give just one example of an oscillator which is
not a self-sustained one.
Counterexample. Consider a famous bob pendulum consisting of a load on a rope,
whose other end is fixed. If we give the load an initial kick, it will start to oscillate,
1 Another popular spelling is “Witt” which is widely used in literature.
2 As explained in “Preface to the second Russian edition” of [14], the name of Vitt was “by
an unfortunate mistake not included on the title page as one of the authors” of [12], but he
has contributed equally with the two other authors.
3 To be more precise, until the power source lasts, as will be explained below; so they are
not perpetuum mobile.
4 In p. 162 of [14] it is said: “The amplitude of these oscillations is determined by the
properties of the system and not by the initial conditions. . . . Whatever the initial conditions,
undamped oscillations are established and (they are) stable.”
5 Except in the case of multistability which will be discussed in Chap. 12. But even then the
number of options is usually quite limited and is anyway offered by the same self-oscillating
system.](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-27-320.jpg)
![14 2 General Remarks
• Any oscillations have power O. Which is the energy per time unit, and monoto-
nously depends on the amplitude A.
• Therefore, in order to maintain non-damped oscillations with a constant ampli-
tude, the system performing them must keep its power at a certain sufficient
level all the time.
• But how can the system do that, if dissipation persistently pumps the power out
of it?
The answer is obvious: The system should simply find the way to feed on some
source of power in order to compensate for its losses.7 Thus we have deduced the
need for the source of power in self-oscillating systems.
Non-linearity
First of all, what is non-linearity? Suppose we have a system about which we would
like to find out whether it is linear or not. Apply some perturbation x1 to it and record
its response y1. Then apply another perturbation x2 and record the response y2 to
that. Then apply perturbation equal to (x1 + x2) and calculate the response y3. Then
calculate the sum (y1 + y2) and compare the two quantities
(y1 + y2) and y3. (2.1)
Are they equal for any chosen x1 and x2? If yes, then the system is linear. If they
are not equal, the system is non-linear. Graphically, linearity can be illustrated as a
straight line on the graph of response y as a function of input x. Anything different
from a straight line would represent a non-linear system.
Let us come back to the system which wants to self-oscillate, i.e., to decide
for itself how to behave and to hold its ground by being resistive to at least minor
influences from the outside world. In order to do this, the system must take power
from the available source in a proper way.
Suppose the system does not oscillate at all, i.e., its amplitude A is zero. At this
state it does not spend power on oscillations, and does not need to compensate for
it. Therefore, the amount of power S taken from the source per time unit should be
zero. They say that the system is in equilibrium. Now let the initial conditions be
such that the amplitude of oscillations is finite A > 0. Then the system is not in
equilibrium and should take power. How?
It is convenient to express powers O and S as functions of A2 rather than A. The
reason is that quite often the power O spent on oscillations is proportional to A2,
i.e., O = kA2 with k being some proportionality constant. Consider this case for
the start.
The amount of power S that enters the system from the source is a function
of A2, and this function can be either linear or non-linear. A few possibilities are
illustrated in Fig. 2.1. In order to maintain oscillations with a certain amplitude A0,
7 In [14] it is said: “A self-oscillating system is an apparatus which produces a periodic
process at the expense of a non-periodic source of energy.”](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-29-320.jpg)
![16 2 General Remarks
Fig. 2.2. Powers in the system as functions of the square of oscillations amplitude A2.
Dashed line: power O spent on oscillations; solid line: power S supplied into the system.
Self-oscillations can occur
2.3.3 Modern Revisions of the Definition of a Self-Sustained System
Andronov et al. [12, 14] have defined a self-sustained system as periodic, but nowa-
days a family of self-oscillations has expanded considerably to include quasiperi-
odic and irregular oscillations, so this requirement is obviously out of date.
Also, the original definition required the self-sustained system to be autonomous,
which in fact means that the power available from the source should be constant and
not depend on time explicitly.8 This definition was revised by Landa [161, 162] in
view of modern developments of oscillations theory. She excluded the word “au-
tonomous” and has thus allowed the source of energy to change in time. This ad-
dition made alone would immediately include oscillations that exist only because
of rhythmic external forcing, i.e., forced oscillations which are not self-sustained.
However, forced oscillations would have the same or similar time scales as the forc-
ing itself. So in order to exclude forced oscillations, Landa adds a requirement that
reads “The complete or partial independence of the frequency spectrum of oscil-
lations from the spectrum of the energy (power) source” [161]. This means that at
least a part of the spectrum of oscillations does not come as a result of the transfor-
mation of the spectrum of the source of power, i.e., the frequency components are
not harmonics or subharmonics of those of the spectrum of the source. At least a
part of the spectrum of oscillations must be defined by the intrinsic properties of the
system itself.
The relaxation of the condition on the constancy of the power source has an
important consequence: it allows one to include into the family of self-sustained
oscillations the ones that are induced merely by random perturbations and would
not occur without them. In Chap. 9 it will be demonstrated that this classification
of noise-induced oscillations is justified, and that they do behave like self-sustained
systems in many respects, and in particular can be synchronized.
8 Although the amount of power actually taken from the source at the given time instant
does depend on the stage of oscillations, and thus depends on time implicitly.](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-31-320.jpg)
![2.4 Duality of the Description of Synchronization 17
2.3.4 Self-Sustained Oscillations and Attractors
A distinctive feature of self-oscillating systems is their ability to self-organize. When
we launch a process in such a system by, e.g., switching it on, the initial conditions
can be chosen at random in a wide range. In general, the time course of a process
thus launched can depend on the initial settings quite substantially. However, a self-
oscillator is very confident about what it is ought to do, and after some transient
(relaxation) time passes by, it arrives at the same regime of oscillations from a large
range of initial conditions. In mathematical terms, such regimes are characterized
by the attractors in the phase space. Sometimes, certain systems can have a choice
of the possible attracting regimes to which they can go, depending on the initial
conditions provided, and this is called multistability. Nevertheless, self-sustained
systems are generally quite firm in their decisions on how to behave, and are resis-
tant to weak attempts to distract them from their course. A mathematical term for
this property is robustness.
2.3.5 Synchronization as a Control Tool
In various applications it might become necessary to amend the conduct of a self-
sustained system either slightly or substantially. One might even want to stop all os-
cillations in it. However, this might not be a straightforward and easy task, given the
above-mentioned stubbornness of self-oscillators. In this respect, our book shows
you the possible ways to control the behavior of self-oscillating systems by means
of clever and inexpensive perturbations. But before one is able to choose the best
way to tame the particular system, it is necessary to classify it, to learn about the
temper and habits of the systems from the given family, and to arm oneself with
the full range of the available taming tools. We wish our reader good luck in this
exciting journey.
2.4 Duality of the Description of Synchronization
Synchronization of oscillations is a phenomenon that was originally discovered by
Christian Huygens in 1665 in a mechanical system: two pendulum “grandfather”
clocks hanging on the same beam [125]. The interaction between the organ pipes
was studied by Rayleigh [243]. The first observations of synchronization in a elec-
tronic tube generators were done by Eccles [58, 75] in relation to the problem of
creating a precision clock and the transmission of naval signals. Almost at the same
time experiments with electric circuits were performed by Appleton [28] and by
van der Pol [292, 293] while they were studying the reception of radio signals with
electric circuits with triodes. The same authors developed the first theoretical ap-
proaches that were able to explain their results to some extent. However, the first
non-linear mathematical theory of synchronization which was able to capture the
phenomena observed much more accurately, was created in the Soviet Union by
Andronov and Witt, also with regard to a very practical problem: stabilization of](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-32-320.jpg)
![18 2 General Remarks
the frequency of a powerful generator of electromagnetic waves by energy-efficient
weak external forcing [13, 299].
In the experiments synchronization was detected by observing Lissajous fig-
ures on the screens of oscilloscopes that provided one with information on the
phase shifts, amplitudes and frequencies. Thus, synchronization can be naturally
described in physical terms like power, frequency or phase. On the other hand, the
systems which can demonstrate synchronization, can be described by non-linear
mathematical equations. Transitions that occur in coupled systems when their pa-
rameters change, can be described in terms of dynamical systems theory including
bifurcation theory. We emphasize that the same phenomena can be described using
different languages, the language of physics or the language of mathematics. But
whatever approach we choose, the underlying phenomena remain the same. In this
book we will analyze the phenomenon of synchronization and the associated effects
using both languages and making a clear connection between these different levels
of description.
2.5 Oscillations Helping Each Other Out
A reader who has reached this point in the book might be already thinking: “First,
they were talking about my heartbeats, whistles, clocks and bottles, then about some
electronic experiments and organ pipes. In between they promised me something
exciting to arise out of the coupling of various devices, and also gave a definition
of some imaginary self-sustained system. These look like all different things to me,
having nothing to do with each other. Even if they are saying that two clocks can be
synchronized, so what? How does it help me to understand what happens to organ
pipes? And above all, what does it have to do with my heart?”
This is a fundamental question which we would be delighted to receive and to
answer.
We need to make a short excursion into the past. Before the beginning of the
20th century, non-linearity was perceived as an annoying misfortune that could be
encountered in this or that physical phenomenon. Every physical problem seemed
to contain some non-linearity, but it would be perceived as its own non-linearity
specific to the given problem [256], just like you might have suggested above. In
the early 1930s Soviet physicist Leonid Mandelstam was the first to recognize the
burning need to develop a unique approach to non-linearity and proposed the ideas
of non-linear thinking. In addition to that, in 1944 in one of his lectures he made
an observation that starting from Kepler laws, most fundamental discoveries made
in physics were in fact oscillatory in this or that way. He also observed that oscilla-
tions were a key element that was common in all traditional subdivisions of physics:
optics, electricity, acoustics, etc. Now we know that oscillations are common in bi-
ology, chemistry, geology, finances and social sciences as well, and this list can
be continued. His ideas of commonness of oscillations and oscillations’ mutual aid
consisted in that there are the same fundamental laws of nature that lie behind os-](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-33-320.jpg)
![2.6 Terms of Bifurcations Theory 19
cillations of all kinds. An understanding of the principles behind oscillations in one
system would help one to understand oscillations in the other systems.
The statement above might not sound immediately obvious, so we continue.
Already at the end of the 19th century it was clear that if one considers small os-
cillations in acoustics and in electricity, and consistently, from the first principles,
derives mathematical equations describing them, the resulting equations will be the
same [243]! Moreover, it was shown that the same equations are valid for small
oscillations in mechanical systems. Is it a coincidence?
Let us go further. Later on, when deriving from the first principles the differ-
ential equations underlying the non-small oscillations in the systems of all kinds
(chemical, biological, physical) it was noticed that quite often these equations ap-
pear equivalent in the sense of topology. The latter means that a change of variables
would reduce one set of equations to another, i.e., that there is no real difference
between them from the viewpoint of mathematics!
At present these ideas are quite well established and might even be occasionally
regarded as trivial. But, as Mandelstam once said “It is the triviality of this, which
is non-trivial” [256]. Thus, the theory of oscillations serves as a common language
that can be spoken by different disciplines. The laws of the theory of oscillations
are common between oscillations of the same class, regardless of the nature of the
particular system demonstrating them.
Coming back to the question in the beginning of this section, we are safe in
saying that all seemingly different systems and phenomena that were mentioned
here, and a huge lot of those not mentioned, just because there are too many of
them, obey the same fundamental principles. If we state that self-oscillations can
be synchronized, this means that all self-oscillations can do that, no matter where
they are found. In the remainder of Part I of this book the simplest paradigmatic
models will be discussed, that describe periodic, chaotic, noisy and noise-induced
oscillations. The mathematical results will predict that certain interesting things can
happen to them. But then qualitatively the same phenomena can occur even in much
more complex systems, provided that their oscillatory properties are equivalent to
those described by simple equations.
Therefore, when learning about, say, phase locking in van der Pol oscillator, one
learns about phase locking in a general periodically self-oscillating system.
2.6 Terms of Bifurcations Theory
In the next chapters we use a number of terms that belong to the theory of differ-
ential equations, including bifurcation theory. It is not possible to give a detailed
introduction into differential equations here, and anyway this is very well done by
other authors before us. Just a few useful sources are [65, 101, 156], and we would
like to mention separately [2] for those who feel that they need to start from a very
basic level. For an excellent historical introduction to dynamical chaos we would
mention [3].](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-34-320.jpg)
![3 1 : 1 Forced Synchronization of Periodic Oscillations
In this chapter we study the simplest case of synchronization: synchronization of
unidirectionally coupled periodic oscillators. Another name for this phenomenon is
forced synchronization, which reflects the fact that one system influences the other,
but does not experience any influence from the other system in return. Another sim-
plifying assumption employed here is that the frequency of external stimulus is suf-
ficiently close to the frequency of natural, i.e., unforced, oscillations. Provided that
the strength and frequency of forcing satisfy certain conditions, a remarkable effect
can take place: the system that experiences only weak external perturbation can start
to oscillate with the frequency equal to the one of this perturbation. They say that
the phenomenon of 1 : 1 phase (frequency) locking, or entrainment, occurs. This is
a special case of a more general phenomenon of n : m synchronization that can be
observed when the forcing frequency ff is not close to the natural frequency f0 of
oscillations in the forced system, but instead is close to a value n
m f0. n : m synchro-
nization will be considered in Chap. 6.
We will start with considering a periodic weakly non-linear oscillator that is
forced harmonically. As a particular example we use a famous paradigm for periodic
self-sustained oscillations, the van der Pol equation, which has been used to describe
a variety of oscillatory phenomena including oscillations of current in electric circuit
[292], signal of electrocardiogram [294], dynamics of semiconductor lasers [49],](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-35-320.jpg)
![22 3 1 : 1 Forced Synchronization of Periodic Oscillations
generation of relativistic magnetrons [168], and the activity of a single neuron [197].
The equation reads
ẍ −
λ − x2
ẋ + ω2
0x = 0. (3.1)
Here, dots over the variables denote derivatives over time t, λ is the non-linearity
parameter and also the bifurcation parameter: at λ 0 there are no self-oscillations,
and the only stable solution of the system is a stable fixed point at the origin. At
λ = 0, Andronov–Hopf bifurcation occurs, as a result of which the fixed point be-
comes unstable, and a stable limit cycle is born. At the moment of birth, oscillations
on the limit cycle are harmonic, and their frequency is exactly equal to ω0 0,
which is also called eigenfrequency. If λ is positive and small, i.e., 0 λ 1,
the periodic self-sustained oscillations remain almost harmonic, and their frequency
remains approximately equal to the value of ω0. The solution to (3.1) for large t,
i.e., after the system has relaxed to the limit cycle from the arbitrarily chosen initial
conditions, can be approximately described by
x(t) = A cos(ω0t + ϕ0), A = const, ω0 = const, ϕ0 = const, (3.2)
where A is the amplitude, ω0 is the frequency, and ϕ0 is the initial phase of oscil-
lations. The respective phase portrait on the plane (ẋ, x) and the realization of x(t)
are shown in Fig. 3.1 by a black line. This is the case of the so-called weakly non-
linear oscillator,1 which can be analyzed analytically by means of the approximate
methods of the theory of oscillations.
Generally, by a weakly non-linear oscillator we understand a system with a limit
cycle, whose control parameters are just above the values corresponding to a super-
critical Andronov–Hopf bifurcation.2 Note that when the non-linearity λ in (3.1) is
no longer small, the oscillations, although remaining periodic, are no longer close
to harmonic, their amplitude grows and the frequency is less than ω0 (Fig. 3.1, grey
line). The larger the λ, the slower the oscillations, and the bigger their amplitude is.
Now, let us introduce external periodic forcing into the system in its simplest
harmonic form as follows:
ẍ −
λ − x2
ẋ + ω2
0x = B cos(Ωt). (3.3)
Here, B and Ω are the strength (amplitude) and frequency of the external forcing,
respectively. The solutions of (3.3) at ω0 = 1, fixed small value of B = 0.01 and
four different values of Ω close to 1 are illustrated in Fig. 3.2. The external forcing
F(t) = B cos(Ωt) (3.4)
is shown by black in the fourth column together with the solution x(t). Note that the
amplitude B of forcing here is much smaller than the amplitude of x. That is why,
in order to allow the reader to compare the details of the behavior of both x and F,
in the last column of Fig. 3.2 we show not F, but 10F.
1 Or nearly sinusoidal, as they are called in [12, 14].
2 There are two forms of Andronov–Hopf bifurcation, a supercritical and a subcritical one.
The former is encountered more often, therefore in what follows we will call it simply
“Andronov–Hopf bifurcation” for brevity.](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-36-320.jpg)
![24 3 1 : 1 Forced Synchronization of Periodic Oscillations
forcing. Note that synchronized oscillations have constant amplitude, and the values
of x at the local maxima are the same from one oscillation to another.
However, when the forcing frequency Ω is not close enough to ω0 (Ω = 0.9 and
Ω = 1.1), i.e., frequency detuning is not small, an interesting phenomenon occurs:
the amplitude of oscillations oscillates itself. This is called beating. The instanta-
neous amplitudes, that are roughly half distances between the closest maxima and
minima x(t), oscillate periodically with a certain beat frequency. To some extent,
this can be visible in the Poincaré section, defined by ẋ = 0, ẍ 0, that shows the
maxima against the values of the forcing F taken at the same instants (Fig. 3.2, third
column). Later we will consider the beat frequency in more detail and make some
theoretical analyses.
Perhaps more importantly, when the amplitude of oscillations is not constant the
forced oscillations are not synchronous with the forcing: when F takes its maximal
values, x can take any value. The projection (F, x) is very informative: it is clearly
visible that x and F move independently of each other. In more rigorous terms,
the oscillations are quasiperiodic: the phase trajectories lie on the two-dimensional
invariant tori whose Poincaré sections are closed curves. This regime corresponds
to the absence of synchronization between the system and the forcing.
In Fig. 3.2 the 1 : 1 synchronization phenomenon is illustrated numerically. How-
ever, for the weakly non-linear oscillator (3.3) considered, synchronization also al-
lows for analytical treatment, which will be illustrated in the next sections.
3.1 Phase of Quasiharmonic Oscillations
Here, we need to introduce an important idea closely associated with the phenom-
enon of synchronization—the idea of phase. When discussing the synchronization
illustrated in Fig. 3.2, we mentioned the “stage” of oscillations, which is the current
position of the system inside the given cycle of oscillations, e.g., the beginning, the
first quarter, the middle, the end, etc. We need a quantity characterizing the “stage”
of oscillations at any given time moment t: call it phase ψ(t). For a purely har-
monic function of time like in (3.2) the phase can be introduced uniquely as an
argument of the cosine or sine, which in the given case will be ψ(t) = ω0t + ϕ0.
In Fig. 3.3(a) a harmonic function cos(1.005t + 0.5) is shown by a solid line that
represents harmonic forcing. The period of the cosine function is 2π, so if we
start to observe the cosine at some time moment t, the onset of the nth full os-
cillation cycle (n = 1, 2, . . .) can be characterized by the values of ψ(t) equal to
ϕ0 +2π(n−1), and the end of it by ϕ0 +2πn. Thus, the first oscillation cycle will be
within ψ(t) ∈ [ϕ0; ϕ0 + 2π], and in terms of time inside the interval t ∈ [0; 2π/ω].
For harmonic oscillations, phase is a linear function of time. In Fig. 3.3(c) the phase
ψ(t) = (1.005t + 0.5) of the signal in (a) is shown with filled circles. In (a) filled
circles indicate the values of the signal at the instants t = 2πn/1.005, i.e., when
phase ψ(t) changes by 2π.
In this chapter we will deal with oscillations in a forced weakly non-linear sys-
tem. Such oscillations, generally not being harmonic, can be viewed as almost har-](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-38-320.jpg)
![26 3 1 : 1 Forced Synchronization of Periodic Oscillations
3.2 Derivation of Truncated Equations for Phase Difference and
Amplitude
It should be noted that the exact oscillating solution of (3.3) for the arbitrary values
of parameters λ, B and Ω cannot be found analytically with the mathematical tools
available so far. However, for a certain range of values of these parameters, one can
analytically find the approximate solutions that would describe the unknown exact
solutions with a certain degree of accuracy. This would be quite sufficient from
the practical viewpoint, since whatever is measured in a real experiment is anyway
measured with a certain error. Hence, approximate theory can be good enough when
one compares it with an experiment. The idea of calculations similar to the ones
presented here first occurred to Andronov and Witt in the 1920s [13, 299]. The
significance of their results is that they were one of the first to successfully analyze
the non-linear systems by the approximate methods, and to provide an accurate
explanation of the earlier experimental observations of synchronization in electric
circuits. The analysis of equations of the form similar to that of (3.3), i.e., of non-
linear, dissipative ordinary differential equations of the second order with weak non-
linearity and with periodic excitation, was also done by Cartwright [61, 62], Gillies
[88], Holmes and Rand [117], Arrowsmith [36]. An introduction into this analysis
was made in [179] and [110] with more references.
In what follows, we will restrict our analysis to the small values of λ, for which
without the forcing (B = 0) the solution is almost harmonic (3.2). The addition
of forcing (B = 0) will obviously change the solution. However, let us assume
that the forcing is not too strong, i.e., B is not large as compared to the amplitude
A0 of unperturbed self-oscillations, and the forcing frequency Ω is only slightly
different from ω0. Then the solution of (3.3) can be approximately described as a
quasiharmonic oscillation, i.e., oscillation in the form of (3.2), whose amplitude A
and the argument ψ (phase) of the cosine are perturbed by the forcing.
Since the system, (3.3), under study is close to being linear with small λ, it
is natural to suppose that its response to an external forcing at the frequency Ω
contains the frequency component Ω. We will thus be looking for a solution in the
form of a quasiharmonic function of time, namely,
x(t) = A(t) cos(Ωt + ϕ(t)). (3.5)
Here, A(t) is the envelope of the oscillations x(t) illustrated by Fig. 3.2 (fourth
column). A does not change in time when synchronization occurs, and oscillates
slowly when beating starts.
We assume that both A(t) and ϕ(t) are slow functions of time compared to the
function cos(Ωt). Mathematically, this condition can be written as
Ȧ(t) ΩA(t), |ϕ̇(t)| Ω. (3.6)
The full phase ψ(t) of the forced oscillations is
ψ(t) = Ωt + ϕ(t), (3.7)](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-40-320.jpg)
![3.2 Derivation of Truncated Equations for Phase Difference and Amplitude 27
while the phase of forcing ψf (t) is
ψf (t) = Ωt.
Hence, ϕ(t) is the phase difference between the forcing and the forced oscillations.
When ϕ(t) is a constant, oscillations x(t) in the system are 1 : 1 synchronized by
the external forcing and are harmonic with frequency Ω. When ϕ(t) changes in
time, there is no 1 : 1 synchronization. Thus, in order to reveal the synchronization
conditions, we need to formulate the explicit equations describing the evolution of
ϕ and A in time. Synchronization will mean that there is/are stable fixed point/s in
these equations, so we will have to find the conditions for these points to exist and
to be stable.
To derive the equations for ϕ and A, one can use the method of averaging, also
known as the Krylov–Bogoliubov4 method [150]. In the following, for brevity we
will omit the brackets “(t)” denoting the explicit dependence on time of A and ϕ. If
we calculate the time derivative of x(t) rigorously, we obtain
ẋ = Ȧ cos(Ωt + ϕ) − AΩ sin(Ωt + ϕ) − Aϕ̇ sin(Ωt + ϕ). (3.8)
By representing the solution in the form of (3.5), instead of one independent phase
variable x(t) we introduce two phase variables: A(t) and ϕ(t). Thus, an ambiguity
is introduced into the system. In order to remove the introduced ambiguity, we have
to specify an additional condition that A(t) and ϕ(t) should satisfy. It is convenient
to set such a condition that the derivative of x(t) is a simple expression in the form
ẋ = −AΩ sin(Ωt + ϕ), (3.9)
which would immediately imply
Ȧ cos(Ωt + ϕ) − Aϕ̇ sin(Ωt + ϕ) = 0. (3.10)
Next, we need to find ẍ. The calculations can be continued using the expressions
above that contain sines and cosines, but it is usually more convenient to operate
with exponential functions. Thus, we want to express all trigonometric functions in
x(t), ẋ and ẍ in terms of exponents of complex arguments. We start from reformu-
lating the solution, (3.5),
x = A cos(Ωt + ϕ) = A
ei(Ωt+ϕ) + e−i(Ωt+ϕ)
2
=
eiΩt Aeiϕ + e−iΩt Ae−iϕ
2
.
Let us introduce a complex function of time a, such that
a = Aeiϕ
, a∗
= Ae−iϕ
, (3.11)
where the asterisk denotes the complex conjugate. Then x(t) can be represented
through a as
4 Bogoliubov is also sometimes spelled as Bogolyubov or Bogolioubov in literature.](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-41-320.jpg)
![3.2 Derivation of Truncated Equations for Phase Difference and Amplitude 29
Regroup terms
iΩȧeiΩt
+
(ω2
0 − Ω2)
2
aeiΩt
+ a∗
e−iΩt
− λ
iΩ
2
aeiΩt
+ λ
iΩ
2
a∗
e−iΩt
+
1
4
a2
ei2Ωt
+ a∗2
e−i2Ωt
+ 2aa∗
iΩ
2
aeiΩt
− a∗
e−iΩt
= B
eiΩt + e−iΩt
2
.
Open the brackets
iΩȧeiΩt
+
(ω2
0 − Ω2)
2
aeiΩt
+ a∗
e−iΩt
− λ
iΩ
2
aeiΩt
+ λ
iΩ
2
a∗
e−iΩt
+
iΩ
8
a3
ei3Ωt
−
iΩ
8
a2
a∗
eiΩt
+
iΩ
8
aa∗2
e−iΩt
−
iΩ
8
a∗3
e−i3Ωt
+
iΩ
4
a2
a∗
eiΩt
−
iΩ
4
aa∗2
e−iΩt
= B
eiΩt + e−iΩt
2
. (3.16)
Collect similar terms and multiply the whole equation by e−iΩt /(iΩ),
ȧ +
(ω2
0 − Ω2)
2iΩ
a + a∗
e−i2Ωt
−
λ
2
a +
λ
2
a∗
e−i2Ωt
+
1
8
a3
ei2Ωt
+
1
8
a2
a∗
−
1
8
aa∗2
e−i2Ωt
−
1
8
a∗3
e−i4Ωt
=
B
2iΩ
1 + e−i2Ωt
. (3.17)
We remind you, that the aim of our calculations is to write down the equations
describing the evolution in time of the complex amplitude a(t), and then to solve
them. Knowing a(t), we will know A and ϕ, and thus we will know the approximate
solution x(t) of van der Pol equation (3.3). However, (3.17) is not simpler than the
original (3.3), and it is not easier to find the amplitude a from it than it was to find x
from (3.3). To simplify the problem we can make more use of the fact of slowness of
A and ϕ and exploit the method of averages by Krylov and Bogoliubov [150]. Note
that a, ȧ and a∗ are slow functions of time as compared to the functions e±nΩt ,
n being an integer number. This means that they almost do not change during one
period of fast oscillations with the frequency Ω. If we average the whole equation
over one period T = 2π/Ω of fast oscillations, we can get rid of the fast terms, and
only the slow terms will remain in the equation. The time average ¯
f T of a smooth
function f (t) over the time interval T is defined as follows:
¯
f T
=
1
T
t0+T
t0
f (t) dt. (3.18)](https://image.slidesharecdn.com/1020377-250521120536-d28cd890/85/Synchronization-From-Simple-To-Complex-1st-Edition-Dr-Alexander-Balanov-43-320.jpg)
Synchronization From Simple To Complex 1st Edition Dr Alexander Balanov
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- 6. Springer Complexity Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems—cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science. Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be suc- cessfully mapped onto quite diverse “real-life” situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications. Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular autom- ata, adaptive systems, genetic algorithms and computational intelligence. The two major book publication platforms of the Springer Complexity program are the mono- graph series “Understanding Complex Systems” focusing on the various applications of complexity, and the “Springer Series in Synergetics”, which is devoted to the quantitative theoretical and method- ological foundations. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works. Editorial and Programme Advisory Board Péter Érdi Center for Complex Systems Studies, Kalamazoo College, USA, and Hungarian Academy of Sciences, Budapest, Hungary Karl J. Friston Institute of Cognitive Neuroscience, University College London, London, UK Hermann Haken Center of Synergetics, University of Stuttgart, Stuttgart, Germany Janusz Kacprzyk System Research, Polish Academy of Sciences, Warsaw, Poland Scott Kelso Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, USA Jürgen Kurths Nonlinear Dynamics Group, University of Potsdam, Potsdam, Germany Linda E. Reichl Center for Complex Quantum Systems, University of Texas, Austin, USA Peter Schuster Theoretical Chemistry and Structural Biology, University of Vienna, Vienna, Austria Frank Schweitzer Systems Design, ETH Zurich, Zurich, Switzerland Didier Sornette Entrepreneurial Risk, ETH Zurich, Zurich, Switzerland
- 7. Springer Series in Synergetics Founding Editor: H. Haken The Springer Series in Synergetics was founded by Herman Haken in 1977. Since then, the series has evolved into a substantial reference library for the quantitative, theoretical and methodological foun- dations of the science of complex systems. Through many enduring classic texts, such as Haken’s Synergetics and Information and Self- Organization, Gardiner’s Handbook of Stochastic Methods, Risken’s The Fokker Planck-Equation or Haake’s Quantum Signatures of Chaos, the series has made, and continues to make, important contri- butions to shaping the foundations of the field. The series publishes monographs and graduate-level textbooks of broad and general interest, with a pronounced emphasis on the physico-mathematical approach.
- 8. Alexander Balanov, Natalia Janson, Dmitry Postnov, Olga Sosnovtseva Synchronization From Simple to Complex With 150 Figures
- 9. Dr. Alexander Balanov Dr. Natalia Janson Loughborough University Loughborough University Department of Physics School of Mathematics Loughborough Ashby Road Leicestershire LE11 3TU Loughborough United Kindom Leicestershire LE11 3TU a.balanov@lboro.ac.uk United Kingdom n.b.janson@lboro.ac.uk Prof. Dmitry Postnov Dr. Olga Sosnovtseva Saratov State University Technical University of Denmark Department of Physics Department of Physics Astrakhanskaya 83 Building 309 Saratov, Russia 410026 2800 Lyngby, Denmark postnov@chaos.ssu.runnet.ru olga@fysik.dtu.dk ISBN 978-3-540-72127-7 e-ISBN 978-3-540-72128-4 DOI 10.1007/978-3-540-72128-4 Library of Congress Control Number: 2008929606 © 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad- casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro- tective laws and regulations and therefore free for general use. Typesetting and Production: VTEX, Vilnius Cover design: WMX Design GmbH, Heidelberg, Germany Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
- 10. To our parents
- 11. Preface This book is written by scientists who live in different countries (United King- dom, Denmark, Russia), but who have graduated from, and were established as researchers at the same place: The Laboratory of Nonlinear Dynamics, Department of Physics, Saratov State University, Russia. Being apart for many years, we have united in one team again to write this book. Why? We aim to summarize both classical results that are crucial for the understanding of the concept of synchronization, and an up-to-date account of the accompanying fascinating phenomena. The main theme that runs throughout the book is that in- teraction between complex systems is governed by the same universal principles. We strive to explain the material in a way that the newcomers to the field would hopefully appreciate, namely, • From simple calculations to advanced theoretical approaches • From simple dynamics to complex behavior • From mathematical and physical to general perspectives Assuming only the basic knowledge of mathematics, our book takes the reader to the frontiers of what is currently known about this research area. The classical approach to synchronization we have learned by heart during our regular and inevitably hot discussions, and most of the results on the new synchro- nization phenomena we obtained together. It is therefore difficult to separate scien- tific contribution and to compare the efforts made by each co-author, so we decided to arrange the list of authors in alphabetic order to emphasize an equal investment of their time, ideas and enthusiasm. This book would not have been possible without the help of many people. First of all, we are deeply indebted to our teacher Prof. V.S. Anishchenko who has intro- duced us to Nonlinear World and who patiently taught us to properly speak the lan- guage of science. We are grateful to our teachers and colleagues Prof. V.V. Astakhov and T.E. Vadivasova for their active support and many invaluable discussions dur- ing the years. We extend our thanks to Prof. E. Mosekilde, Prof. P. McClintock, Prof. S.K. Han, who are our closest collaborators in the field of synchronization, and to Prof. N.-H. Holstein-Rathlou, Prof. D. Marsh, and Prof. H. Braun, with whom we have been enjoying collaborations in the field of modeling of biological sys- tems. Our special thanks are due to Prof. E. Schöll who has encouraged us to write this book. We acknowledge fruitful discussions with our colleagues A. Pikovsky, M. Rosenblum, M. Zaks, J. Kurths, L. Schimansky-Geier, A. Neiman, A. Nikitin,
- 12. viii Preface and A. Silchenko on various aspects of synchronization. We gratefully acknowl- edge the help of S. Malova with references and of P. Sherbakov with experiments. We would also like to warmly thank Victoria Sosnovtseva for making funny illustra- tions especially for this book. Finally, we would like to express our sincere gratitude to our families for their constant support and inspiration. Over the years our studies were supported by the Russian Foundation for Ba- sic Research (Russia), the U.S. Civilian Research and Development Foundation (USA), Engineering and Physical Sciences Research Council (UK), Medical Re- search Council (UK), The Leverhulme Trust (UK), Forskningsrådet for Natur og Univers (Denmark), and the European Union through the Network of Excellence BioSim. Loughborough, Alexander Balanov Saratov, Natalia Janson Lyngby, Dmitry Postnov May 2008 Olga Sosnovtseva
- 13. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part I General Mechanisms of Synchronization 2 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 What Are We Going to Talk About? . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Topics to Consider. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Self-Sustained Oscillations: A Key Concept in Synchronization Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.1 Features of Self-Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Features of Self-Oscillating Systems . . . . . . . . . . . . . . . . . . . . 13 2.3.3 Modern Revisions of the Definition of a Self-Sustained System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.4 Self-Sustained Oscillations and Attractors . . . . . . . . . . . . . . . 17 2.3.5 Synchronization as a Control Tool . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Duality of the Description of Synchronization . . . . . . . . . . . . . . . . . . . 17 2.5 Oscillations Helping Each Other Out . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.6 Terms of Bifurcations Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 1 : 1 Forced Synchronization of Periodic Oscillations . . . . . . . . . . . . . . . 21 3.1 Phase of Quasiharmonic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Derivation of Truncated Equations for Phase Difference and Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Amplitude of Unperturbed Oscillations at Small Non-linearity . . . . . 31 3.4 Analysis of Truncated Equations for Weak Forcing . . . . . . . . . . . . . . 32 3.5 Derivation of Truncated Equations in Descartes Coordinates. . . . . . . 34 3.6 Analysis of Truncated Equations in Descartes Coordinates . . . . . . . . 37 3.7 Synchronization Region from the Truncated Equations: Non-bifurcational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.8 Fourier Power Spectra at Strong Forcing . . . . . . . . . . . . . . . . . . . . . . . 50 3.9 Phase Locking and Suppression: Numerical Simulation . . . . . . . . . . . 56
- 14. x Contents 3.9.1 Phase Locking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.9.2 Suppression of Natural Dynamics . . . . . . . . . . . . . . . . . . . . . . 60 3.10 Phase Locking and Suppression: Experiment . . . . . . . . . . . . . . . . . . . 62 3.10.1 Amplitudes from Oscilloscopes . . . . . . . . . . . . . . . . . . . . . . . . 64 3.11 Beat Frequency: Theory, Simulations and Experiment . . . . . . . . . . . . 67 3.11.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.11.2 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.11.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4 1 : 1 Mutual Synchronization of Periodic Oscillations . . . . . . . . . . . . . . . 75 4.1 Truncated Equations for Weakly Non-linear Oscillators . . . . . . . . . . . 77 4.2 Periodic Oscillators with Dissipative Coupling . . . . . . . . . . . . . . . . . . 80 4.2.1 Symmetric Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.2 Asymmetric Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2.3 Oscillation Death . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3 Dissipative Coupling: Numerical Simulation . . . . . . . . . . . . . . . . . . . . 84 4.3.1 Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3.2 Bifurcations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3.3 Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.4 Reactive Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.1 Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4.2 Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.4.3 Bifurcations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.4.4 Phase Multistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Reactive Coupling and the Saddle Torus . . . . . . . . . . . . . . . . . . . . . . . 95 4.5.1 Hypothesized Structure of the Phase Space . . . . . . . . . . . . . . . 96 4.6 Generality of Bifurcational Transitions at Reactive Coupling . . . . . . 97 4.7 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.7.1 Phase Locking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.7.2 Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.8 Comparison of Synchronization Transitions in Forced and in Mutually Coupled Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5 Homoclinic Mechanism of Synchronization of Periodic Oscillations . . 105 5.1 Global Bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.1.1 Features of a Homoclinic Bifurcation of a Cycle . . . . . . . . . . 110 5.2 Homoclinics Inside Synchronization Tongue? . . . . . . . . . . . . . . . . . . . 111 5.3 How Homoclinics Leads to Synchronization . . . . . . . . . . . . . . . . . . . . 114 5.4 Synchronization in a Bacteria–Viruses Model . . . . . . . . . . . . . . . . . . . 117 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6 n : m Synchronization of Periodic Oscillations . . . . . . . . . . . . . . . . . . . . . 121 6.1 Important Definitions Relevant to n : m Synchronization . . . . . . . . . . 121 6.1.1 Poincaré Return Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.1.2 Phase of Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
- 15. Contents xi 6.1.3 Phase of Oscillations via Poincaré Section . . . . . . . . . . . . . . . 122 6.1.4 Poincaré Winding (Rotation) Number . . . . . . . . . . . . . . . . . . . 123 6.1.5 Synchronization Order n : m . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.2 1 : 1 Forced Synchronization in Weakly Non-linear Oscillators . . . . . 123 6.2.1 3 : 1 Phase (Frequency) Locking . . . . . . . . . . . . . . . . . . . . . . . . 128 6.2.2 3 : 1 Suppression of Natural Dynamics . . . . . . . . . . . . . . . . . . 131 6.3 n : m Synchronization in Strongly Non-linear Oscillators with Spiky Forcing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.3.1 2 : 3 Phase (Frequency) Locking . . . . . . . . . . . . . . . . . . . . . . . . 136 6.3.2 The Route to 2 : 3 Suppression . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.4 Circle Map: Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.4.1 Amplitude and Phase of Oscillations . . . . . . . . . . . . . . . . . . . . 139 6.4.2 From Differential to Discrete Equation for Phase . . . . . . . . . . 141 6.5 Circle Map: Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.6 Arnold Tongues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.7 n : m Synchronization: Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7 1 : 1 Forced Synchronization of Periodic Oscillations in the Presence of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.1 Introductory Comments on Random Processes . . . . . . . . . . . . . . . . . . 150 7.1.1 One-Dimensional Probability Density, Mean and Variance . . 150 7.1.2 Two-Dimensional Probability Density, Correlation and Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.1.3 Stationary Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.1.4 Correlation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.1.5 Correlation Between Two Different Processes . . . . . . . . . . . . 155 7.1.6 Spectrum of a Wide-Sense Stationary Process . . . . . . . . . . . . 156 7.2 Truncated Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.3 Simplification of the Fluctuational Terms in Truncated Equations . . 158 7.4 Probability Density Distribution of the Phase Difference . . . . . . . . . . 165 7.4.1 Case of Q > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.5 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.6 Probability Density Distribution of the Phase Difference, Continued 172 7.7 Mean Frequency of Forced Oscillations with Noise . . . . . . . . . . . . . . 174 7.8 Interpretation of Phase Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.9 Phase Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.10 Full-Scale Biological Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.11 Effects of Noise on the Spectrum of a Synchronized System . . . . . . . 185 7.11.1 Effect of Noise on the Spectrum of Oscillations Synchronized by Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . 189
- 16. xii Contents 8 Chaos Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 8.1 What Is Chaos? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 8.1.1 Exponential Divergence of Phase Trajectories . . . . . . . . . . . . 192 8.1.2 Chaos Properties in Terms of Phase Space . . . . . . . . . . . . . . . 193 8.1.3 Chaos Properties in Terms of Spectra . . . . . . . . . . . . . . . . . . . 197 8.2 What Does Synchronization of Chaos Encompass? . . . . . . . . . . . . . . 197 8.2.1 Chaos Synchronization: Different Manifestations . . . . . . . . . 197 8.2.2 Chaos Synchronization in a Classical Sense . . . . . . . . . . . . . . 198 8.3 Phase and Basic Frequency of Chaotic Oscillations . . . . . . . . . . . . . . 199 8.4 Forcing Chaos Periodically: What to Expect? . . . . . . . . . . . . . . . . . . . 201 8.4.1 Phase Locking of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.4.2 Suppression of Chaos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.4.3 Any Other Options? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.4.4 Interacting Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.5 Synchronization of Chaos by Periodic Forcing . . . . . . . . . . . . . . . . . . 205 8.5.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.5.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8.6 Synchronization of Periodic Oscillations by Chaos . . . . . . . . . . . . . . . 212 8.6.1 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.6.2 Poincaré Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 8.6.3 Phase Difference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 8.6.4 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.7 Mutual Synchronization of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 8.7.1 Phase/Frequency Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 8.7.2 Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 8.7.3 Phase Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 8.8 Homoclinic Synchronization of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.9 Effects of Noise on a Synchronized Chaos . . . . . . . . . . . . . . . . . . . . . . 232 8.9.1 Chaotic System Frequency-Locked by a Harmonic Signal . . 233 8.9.2 Periodic System Suppressed by Chaotic Forcing . . . . . . . . . . 237 8.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 9 Synchronization of Noise-Induced Oscillations . . . . . . . . . . . . . . . . . . . . 239 Stochastic Limit Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.1 Noise-Induced Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 9.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 9.2.1 Morris–Lecar Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 9.2.2 Monovibrator Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 9.3 Coherence Resonance Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 9.4 Frequency and Phase Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 9.4.1 Frequency Locking: Electronic Experiment . . . . . . . . . . . . . . 249 9.4.2 Phase Locking: Coupled Morris–Lecar Models . . . . . . . . . . . 251 9.4.3 Phase Dynamics Inside the Synchronization Region: Electronic Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 9.5 Synchronization via Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
- 17. Contents xiii 10 Conclusions to Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Part II Case Studies in Synchronization 11 Synchronization of Anisochronous Oscillators . . . . . . . . . . . . . . . . . . . . . 265 11.1 Phase Velocity Field and Coupling Vector . . . . . . . . . . . . . . . . . . . . . . 266 11.2 Effective Coupling Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 11.2.1 Asymptotic Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 11.2.2 Effective Coupling Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 11.3 Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 11.4 Examples of 2D Anisochronous Oscillators . . . . . . . . . . . . . . . . . . . . . 273 11.5 Synchronization near the Homoclinic Bifurcation . . . . . . . . . . . . . . . . 279 11.5.1 Weak Coupling Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 11.5.2 Finite Coupling Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 11.5.3 Strong Coupling with Moderate μ . . . . . . . . . . . . . . . . . . . . . . 288 11.5.4 Summary on Synchronization near Homoclinic Bifurcation . 289 11.6 Phase Locking Patterns of Coupled Fast-and-Slow Oscillators . . . . . 290 11.6.1 Antiphase Locking in Coupled FitzHugh–Nagumo Models . 290 11.6.2 Out-of-phase Synchronization via Slow Channels . . . . . . . . . 293 11.7 Synchronous Patterns in Coupled Morris–Lecar Models . . . . . . . . . . 296 11.7.1 Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 11.7.2 Overview of the Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 11.7.3 Structure of Arnold Tongue for Antiphase Solution . . . . . . . . 300 Chaotic Bursting and Torus Breakdown. . . . . . . . . . . . . . . . . . 306 11.7.4 Crises at the Boundary of Quasiperiodic Regions. . . . . . . . . . 308 11.7.5 Transition to In-phase Synchronization . . . . . . . . . . . . . . . . . . 311 11.7.6 Mechanism of Torus Folding in the Vicinity of Unstable Orbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 11.7.7 Remarks on Synchronization in Morris–Lecar Systems. . . . . 314 11.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 12 Phase Multistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 12.1 Period-Doubling Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 12.1.1 Dynamics of Coupled Rössler Systems . . . . . . . . . . . . . . . . . . 320 12.1.2 Mapping Approach to Multistability . . . . . . . . . . . . . . . . . . . . 330 12.2 Self-Modulated Oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 12.2.1 Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 12.2.2 Phase Dynamics of Coupled Oscillators . . . . . . . . . . . . . . . . . 337 12.3 Bursting Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 12.3.1 Simple Qualitative Approach to Phase Multistability . . . . . . . 342 12.3.2 Dynamics of Coupled Bursters . . . . . . . . . . . . . . . . . . . . . . . . . 344 12.3.3 Multistability Induced by Dephasing . . . . . . . . . . . . . . . . . . . . 349 12.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
- 18. xiv Contents 13 Synchronization in Systems with Complex Multimode Dynamics . . . . 353 13.1 Synchronization of Chaotic Systems with Fast and Slow Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 13.1.1 Single System with Two Time Scales . . . . . . . . . . . . . . . . . . . 355 13.1.2 Coupled Systems with Two Mode Dynamics . . . . . . . . . . . . . 360 13.1.3 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 13.2 Generation and Synchronization of Oscillations with Several Noise-Induced Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 13.2.1 Description of Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 13.2.2 Characterizing Collective Response by Spectra . . . . . . . . . . . 364 13.2.3 Mutually Coupled Excitable Units . . . . . . . . . . . . . . . . . . . . . . 365 13.2.4 Three Coupled Excitable Units . . . . . . . . . . . . . . . . . . . . . . . . . 369 13.2.5 Two Mutually Coupled Excitable Units with Inhibitory Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 13.3 Synchronization of Chaotic Systems with Denumerable Set of Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 13.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 14 Synchronization of Systems with Resource Mediated Coupling . . . . . . 377 14.1 Neural Synchronization via Potassium Signaling . . . . . . . . . . . . . . . . 379 14.1.1 Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 14.1.2 Identical Cells: Competing In-phase and Antiphase Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 14.1.3 Heterogeneous Cells: Dynamical Patterns . . . . . . . . . . . . . . . . 386 14.2 Multimode Dynamics in Linear Array of Electronic Oscillators . . . . 388 14.2.1 Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 14.2.2 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 14.2.3 Intracluster Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 14.3 Cascaded Microbiological Oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . 395 14.3.1 Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 14.3.2 Spatial Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 14.4 Synchronization Patterns in Kidney Autoregulation . . . . . . . . . . . . . . 401 14.4.1 Vascular-Nephron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 14.4.2 Coupling-Induced Inhomogeneity . . . . . . . . . . . . . . . . . . . . . . 405 14.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 15 Conclusions to Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 And finally... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
- 19. 1 Introduction It would not be too much of an exaggeration to say that oscillations are one of the main forms of motion. They range from the periodic motion of planets to random openings of ion channels in cell membranes. They are observed at various levels of organization, have various origins and various properties. Since Newton’s crack at the three-body problem and until just a few decades ago, the range of phenomena regarded as oscillations were limited to damped, periodic and quasiperiodic oscil- lations at best. A significant achievement of the second half of the 20th century is the admission of deterministic chaos and noise-induced rhythms as equals into the oscillation family. Nature is not based on isolated individual systems. It is rich in connections, in- teractions and communications of different kinds that are complex beyond belief. With this, synchronization is the most fundamental phenomenon associated with oscillations. It is a direct and widely spread consequence of the interaction of dif- ferent systems with each other. In most general terms, synchronization means that different systems adjust the time scales of their oscillations due to interaction, but there is a large variety of its manifestations and of the accompanying fascinating phenomena. Anyone writing a book on synchronization is faced with two problems: on one hand, one has to deal with a huge amount of material on the particular aspects and effects; and on the other hand, there is a need to formulate a universal approach that would embrace all the particular cases. Fortunately, an essential contribution to the second problem has been made by Pikovsky, Rosenblum, and Kurths in their recent book [214], that has provided a contemporary view on synchronization as a univer- sal phenomenon that manifests itself in the entrainment of rhythms of interacting self-sustained systems. This viewpoint is in agreement with the approach developed since the time of Huygens, and is completely shared by ourselves. In writing the present book we were motivated by the following considerations: • Recently, a large variety of new synchronization phenomena were discovered that are inherent in complex (chaotic) systems, but do not occur in simple peri- odic oscillators. With the modern fascination for the beauty and the complexity of the new effects, there is a tendency to forget about the basic phenomena and theoretical results associated with “simply” periodic oscillations. This is largely due to the fact that not all involved in the studies of these phenomena, and espe- cially younger researchers and students, have the respective education. It turns
- 20. 2 1 Introduction out to be difficult to recommend a book, which would consistently present, equa- tion after equation, the most fundamental theoretical results on synchronization. Without such background, it is problematic to analyze the synchronization of ir- regular oscillations from the general viewpoint, and to avoid discovering “new” effects that often appear to be merely manifestations of the general principles in a particular situation. • There is a number of fascinating aspects of synchronization (phase multistabil- ity, dephasing, self-modulation, etc.), that are observed in a variety of systems and with various types of interaction, that have not been discussed yet in the framework of the general concept of synchronization. In order to cover the above problems, our book contains two parts. The first part is a consistent and detailed description of the classical approach to forced and mutual synchronization that is based on frequency/phase locking and suppression of nat- ural dynamics. It is oriented to the people not familiar with the fundamental results of synchronization theory obtained by a number of physicists and mathematicians, such as B. van der Pol, A.A. Andronov, A.A. Vitt, M.L. Cartwright, A.W. Gillies, P.J. Holmes, D.A. Rand, R.L. Stratonovich, V.I. Tikhonov, P.S. Landa, D.G. Aron- son and co-authors, and published in their original works. It was our aim: • To reproduce in every detail the derivations of the most fundamental results, which until now were given only schematically and presented a significant chal- lenge for beginners because of the traditional brevity typical of the scientific works of the beginning and middle of the 20th century. We have made every effort to make the reading easy for non-experts, to reduce to the minimum the need to refer to other literature when following the calculations or the descrip- tion of geometrical effects, and to exclude expressions like “It is easy to show.” As a result, the lengths of the respective sections have increased substantially as compared to those in the original books and papers, but we believe it was worth doing this and hope that the readers will find this material helpful. • To describe the same phenomena using different languages: the ones of physics and of mathematics. In the early experiments on synchronization, the latter was detected by means of listening to the volume of sound (organ pipes), visually observing the positions of pendulums (clocks), and later Lissajous figures and Fourier power spectra on the oscilloscopes (electric circuits). Thus, synchro- nization can be naturally understood in physical terms like power, frequency or phase. On the other hand, the systems that synchronize can be described by non-linear mathematical equations. Transitions that occur in coupled systems when their parameters change, can be described in mathematical terms of bi- furcation and stability theory. In this book we will analyze the phenomena of synchronization and the associated effects using both languages and making a clear connection between these different means of description. • To generalize theoretical results to complex oscillations. An important achieve- ment of modern oscillations theory is the recognition of the role of irregular oscillations that can be either deterministic or stochastic. We start by consid- ering synchronization in simple periodic oscillators. Then we move to chaotic
- 21. 1 Introduction 3 and stochastic oscillations and show that in spite of their complexity, they can synchronize according to the same mechanisms as periodic ones. We will deem to have achieved our goal, if after reading this part the reader will be convinced that very different types of oscillations obey the same mechanisms of synchronization, although the particular manifestations can be different. The second part is devoted to the general mechanisms and principles of synchro- nization, describing them with regard to the non-linear properties of the particular classes of systems and couplings. We discuss synchronization of anisochronous os- cillations, when fast and slow motions along the trajectory give arise to additional phase-shifted coexisting regimes and thus change the bifurcational structure of the synchronization region. A separate chapter is devoted to the concept of phase mul- tistability and its development in the systems that oscillate with complex waveform (essential for period-doubling and self-modulated oscillations) and have a particular structure of their phase space. The latter might include regions of fast and slow mo- tion, closeness of the trajectories to some singular points, etc. (essential for bursting behavior). The concept of synchronization is extended to the systems with several time scales of either deterministic, or stochastic origin. Finally, we consider coop- erative behavior of systems with a particular type of coupling through the primary resource supply and discuss their applications.
- 22. Part I General Mechanisms of Synchronization
- 23. 7 “Begin at the beginning,” the King said, very gravely, “and go on till you come to the end: then stop.” Lewis Carroll, “Alice in Wonderland” You have to learn the rules of the game. And then you have to play better than anyone else. Albert Einstein This part offers a tutorial description of the mechanisms of synchronization. We start from the beginning: periodic oscillations and analytical approaches. Then we proceed with irregular oscillations, either chaotic or stochastic, and generalize the classical results. Then we stop.
- 24. 2 General Remarks 2.1 What Are We Going to Talk About? “Synchronization, of course, but what is it and why should I bother?” you might ask. Look, everything around us is moving. As René Descartes used to say [71], “Give me the matter and motion and I will construct the universe.” Others say “mo- tion is the mode of existence of matter” [274]. How exactly is the matter moving? One very popular possibility is the motion that demonstrates a certain degree of rep- etition, this would be an oscillation. Your heart is an oscillator, can you hear how it beats? Now consider several oscillators and let them feel each other’s motion, no matter how exactly—the scientists would say “couple them.” Most likely, the coupling will not go unnoticed by any of these systems: all of them will change their behavior to this or that extent. In fact, this is going on in your body right now: you inhale and exhale repetitively, and thus influence the way your heart beats without knowing it perhaps. The basic features of oscillations are their amplitude and shape, but when we talk about repetition of anything, a natural question is “how often?” With this question arises the concept of a characteristic time scale of oscillations. What does coupling have to do with all this? Well, because of coupling all aspects of the system’s behavior would generally change, sometimes most dras- tically. So, before you couple anything that oscillates, it would be good to know the possible consequences in advance, wouldn’t it? We can tell you right now that a lot of things can happen. For example, oscillations can stop altogether, which might be good sometimes, but occasionally disastrous. Or they could become to- tally unpredictable—but you might like it nevertheless because it looks beautiful. But the phenomenon which is most often associated with synchronization is the change of the time scales of interacting systems: if you couple the systems cleverly, they can start to oscillate “syn-chronously,” which means “sharing the same time” [214]. For your heart and breathing this can mean that, say, while you are breathing once, your heart makes three beats exactly. One can say that synchronization is the most fundamental phenomenon that oc- curs in oscillating processes. In most general terms, synchronization can be defined
- 25. 10 2 General Remarks as follows: Synchronization is an adjustment of the time scales of oscillations due to interaction between the oscillating processes. 2.2 Topics to Consider In Part I we consider the general types of non-damped oscillations that can occur in real-life systems, and introduce the three mechanisms by which all of them can be synchronized. More precisely: • In Chap. 3 we describe the phenomenon of 1 : 1 forced synchronization which can occur in self-sustained periodic oscillators, i.e., in systems that, without be- ing influenced externally, demonstrate purely periodic oscillations—a definition of these systems is given in Sect. 2.3. Real-life examples of such systems are clocks, either mechanical or electronic, generators of electromagnetic waves, drills, metronomes, a string of a violin while being bowed, etc. If periodic forc- ing is applied to such systems, and if the frequency of forcing is close to, but slightly different from, the frequency of self-oscillations, the forcing can entrain both their frequency and phase. Two classical mechanisms of forced synchro- nization are introduced: phase (frequency) locking and suppression of natural dynamics. • In Chap. 4 the interaction between two periodic oscillators is described, which are coupled to each other bidirectionally or mutually. If the frequencies of un- coupled oscillators are sufficiently close, then depending on the kind of coupling between them, a number of phenomena can occur. One possibility is 1 : 1 mu- tual synchronization, when both subsystems start to oscillate periodically with the same frequency which is not equal to either of their natural frequencies. Another kind of response induced by coupling is the simultaneous death of os- cillations in both subsystems. Also, one can expect the more complicated phe- nomenon of phase multistability: one out of two (or even out of a larger number) oscillating patterns can be realized at exactly the same set of control parameters, depending on the choice of the initial conditions. However, the mechanisms of synchronization in mutually coupled weakly non-linear oscillators of general type are the same as in forced systems, namely, phase (frequency) locking and suppression. • Chapter 5 considers the third mechanism of synchronization of periodic oscilla- tions via homoclinic bifurcation, which is different from locking or suppression, and which involves global restructuring of the phase space of interacting sys- tems. This mechanism is less general, but nevertheless can be expected in quite a large class of self-oscillators whose autonomous oscillations are highly inho- mogeneous in time. The examples of such systems are populations of microor- ganisms, neuron systems, lasers, etc. We give mathematical definitions that are essential for the description of this synchronization scenario; discuss in detail
- 26. 2.2 Topics to Consider 11 the changes in the phase space that accompany the onset of synchronization via homoclinic bifurcation, and reveal the phenomena associated with this mecha- nism. • Chapter 6 is devoted to a more generic case of n : m synchronization, when, as a result of interaction, the ratio of the time scales of the coupled systems becomes equal to n : m, where n and m are arbitrary integers. Such a situation typically occurs when the natural time scales of the interacting systems are not close to each other. We describe how the main synchronization mechanisms for this particular case are realized, and derive a simple discrete map called the circle map to analyze this type of synchronization. We illustrate n : m synchronization on an example of cardiorespiratory interaction in humans. • In Chap. 7 we discuss the general effects of noise on synchronization of pe- riodic oscillations. We demonstrate that noise, which is inevitably present in all real systems, can evoke very non-trivial phenomena in the dynamics of syn- chronized self-oscillators. Different theoretical approaches for the description of noise-induced phenomena are discussed. For an example of forced 1 : 1 synchro- nization, we analytically study phase and frequency properties of synchronous oscillations in the presence of noise. The theoretical results are illustrated by experiments with electronic self-oscillators and with the cardiovascular system of humans. • Chapter 8 describes the mechanisms of synchronization of chaotic oscillations. The latter was found to be typical dynamical regimes in many real systems. Examples of systems with chaotic dynamics are fluid and gas flows, electri- cal circuits, semiconductors devices, populations of animals, biological objects, and many others. The chapter starts with an explanation of the origin of the dynamical chaos. We discuss different manifestations of synchronization of ir- regular chaotic oscillations. The concept of phase for a non-periodic process is introduced. We describe the synchronization of chaos in terms of phases and frequencies of chaotic oscillations, and also in terms of saddle periodic orbits embedded into chaotic attractors. Forced and mutual synchronization of chaos is discussed. The main mechanisms of chaos synchronization are revealed, and the effects of noise on them are considered. Some results are illustrated by ex- periments with an electronic circuit. • In Chap. 9 synchronization is considered in systems where oscillations are in- duced merely by external random fluctuations. We discuss different classes of dynamical systems where noise alone is able to induce highly regular oscilla- tions with the properties similar to the properties of deterministic self-oscilla- tions. We show that the mechanisms of synchronization characteristic of purely deterministic systems are also valid for noise-induced oscillations. We discuss the peculiarities of synchronization in stochastic systems and illustrate these re- sults on electronic circuits and on the models of neurons.
- 27. 12 2 General Remarks 2.3 Self-Sustained Oscillations: A Key Concept in Synchronization Theory Before we start talking about any synchronization at all, we need to outline more precisely the class of systems and processes in which we can expect it to occur. Systems that oscillate in principle are usually called oscillators. But the systems we are interested in should be capable of demonstrating oscillations that are self- sustained, or self-oscillations. The concept of self-oscillations was first proposed by Andronov, Khaikin and Vitt1 in 1937 [12] (for the English version see [14]2). Self- oscillations form a special, but rather broad class of all oscillating processes and are characterized by the following features. 2.3.1 Features of Self-Oscillations Below we list the features of self-oscillations. • First and foremost, they do not damp, i.e., the repetitive motion of the system does not stop with the course of time, and does not show the tendency to stop.3 • Second and equally important, they oscillate “by themselves,” i.e., not because they are repetitively kicked from outside. • The third feature is perhaps the most intriguing and fascinating: the shape, am- plitude and time scale of these oscillations are chosen by the oscillating system alone.4 An outsider cannot easily change them, e.g., by setting different initial conditions.5 Examples of self-oscillators are a grandfather pendulum clock, a whistle, your throat when you sing a musical note, as well as many musical instruments, your heart and many other biological systems, a bottle of water with a narrow neck that is put vertically with its neck down (water will come out in pulses). In order to prevent possible confusion, we would like to give just one example of an oscillator which is not a self-sustained one. Counterexample. Consider a famous bob pendulum consisting of a load on a rope, whose other end is fixed. If we give the load an initial kick, it will start to oscillate, 1 Another popular spelling is “Witt” which is widely used in literature. 2 As explained in “Preface to the second Russian edition” of [14], the name of Vitt was “by an unfortunate mistake not included on the title page as one of the authors” of [12], but he has contributed equally with the two other authors. 3 To be more precise, until the power source lasts, as will be explained below; so they are not perpetuum mobile. 4 In p. 162 of [14] it is said: “The amplitude of these oscillations is determined by the properties of the system and not by the initial conditions. . . . Whatever the initial conditions, undamped oscillations are established and (they are) stable.” 5 Except in the case of multistability which will be discussed in Chap. 12. But even then the number of options is usually quite limited and is anyway offered by the same self-oscillating system.
- 28. 2.3 Self-Sustained Oscillations: A Key Concept in Synchronization Theory 13 but if we leave it alone, the oscillations will decay and eventually stop due to friction of the whole construction with air, and also at the point of the rope attachment. Of course, a repetitive kicking will resume the oscillations of the pendulum, but these will not be self-sustained because they would damp without the kicks. What if there were no friction in the system? Then the oscillations would not damp, but would that make them self-sustained? No, because the properties of these oscillations would be completely defined by the direction and strength of the initial kick made by an outsider who would wish to launch them: the harder one kicks, the larger the swing will be. This would contradict the third feature of self-oscillations. 2.3.2 Features of Self-Oscillating Systems For self-oscillations to occur, the oscillating system must be designed in a special way—which is quite a popular design, we haste to say. The following three features of the self-oscillating systems are most essential: they must be non-linear systems, there must be dissipation in them, and there must be a source of power. Dissipation Dissipation is a mechanism due to which energy is being lost by the system while it changes its state, i.e., performs a motion. It has to be said that most macroscopic systems are dissipative anyway, since there is always some sort of friction in it. For example, mechanical systems lose energy because their details experience fric- tion with other details or surrounding air. In electronic systems elementary particles bump into other particles, the elements of the circuits heat and thus lose energy. This list can be continued, but the main idea is clear: dissipation is everywhere. It would be pertinent to emphasize again that the systems without (or almost without) dissipation are not self-oscillators. The oscillations in such systems are usually associated with the motion of either very small (microscopic) particles like electrons in an atom, or of very large (megascopic) objects like stars and planets. They do oscillate (rotate around their centers) eternally, but just because the energy of their oscillations is not wasted on friction. Power Source Having established that dissipation is ubiquitous, a natural line of thought occurs: • Oscillations have amplitude A. Which, roughly speaking, is half the difference between the maximal and the minimal values of an oscillating quantity. Note that if oscillations are decaying, their amplitude decreases. If the oscillations are on the contrary expanding, A grows with the time course. For self-oscillations the amplitude A should not change in time.6 6 At least if the oscillations are periodic. If self-oscillations are not periodic, their amplitude will itself oscillate around some average value, neither growing unboundedly, nor tending to zero, like, e.g., in chaotic oscillations described in Chap. 8.
- 29. 14 2 General Remarks • Any oscillations have power O. Which is the energy per time unit, and monoto- nously depends on the amplitude A. • Therefore, in order to maintain non-damped oscillations with a constant ampli- tude, the system performing them must keep its power at a certain sufficient level all the time. • But how can the system do that, if dissipation persistently pumps the power out of it? The answer is obvious: The system should simply find the way to feed on some source of power in order to compensate for its losses.7 Thus we have deduced the need for the source of power in self-oscillating systems. Non-linearity First of all, what is non-linearity? Suppose we have a system about which we would like to find out whether it is linear or not. Apply some perturbation x1 to it and record its response y1. Then apply another perturbation x2 and record the response y2 to that. Then apply perturbation equal to (x1 + x2) and calculate the response y3. Then calculate the sum (y1 + y2) and compare the two quantities (y1 + y2) and y3. (2.1) Are they equal for any chosen x1 and x2? If yes, then the system is linear. If they are not equal, the system is non-linear. Graphically, linearity can be illustrated as a straight line on the graph of response y as a function of input x. Anything different from a straight line would represent a non-linear system. Let us come back to the system which wants to self-oscillate, i.e., to decide for itself how to behave and to hold its ground by being resistive to at least minor influences from the outside world. In order to do this, the system must take power from the available source in a proper way. Suppose the system does not oscillate at all, i.e., its amplitude A is zero. At this state it does not spend power on oscillations, and does not need to compensate for it. Therefore, the amount of power S taken from the source per time unit should be zero. They say that the system is in equilibrium. Now let the initial conditions be such that the amplitude of oscillations is finite A > 0. Then the system is not in equilibrium and should take power. How? It is convenient to express powers O and S as functions of A2 rather than A. The reason is that quite often the power O spent on oscillations is proportional to A2, i.e., O = kA2 with k being some proportionality constant. Consider this case for the start. The amount of power S that enters the system from the source is a function of A2, and this function can be either linear or non-linear. A few possibilities are illustrated in Fig. 2.1. In order to maintain oscillations with a certain amplitude A0, 7 In [14] it is said: “A self-oscillating system is an apparatus which produces a periodic process at the expense of a non-periodic source of energy.”
- 30. 2.3 Self-Sustained Oscillations: A Key Concept in Synchronization Theory 15 Fig. 2.1. Powers in the system as functions of the square of oscillations amplitude A2. Dashed line: power O spent on oscillations; solid line: power S supplied into the system. The approx- imation O = kA2 is used. a S is a linear function of A2, no non-damped oscillations can occur; b S is a non-linear function of A2, oscillations with A0 are stable; c S is a non-linear function of A2, oscillations with A0 are unstable the supplied power S must compensate the dissipated power O. If S is a linear func- tion of A2 as shown in (a), then S and O can intersect only at one zero point, which is equivalent to the absence of oscillations. If S is non-linear as illustrated in (b), then two intersections are possible: at zero and at a certain A2 0. This means that if the amplitude A of oscillations reaches the value of A0, the lost power is being com- pensated. With this, if A > A0, the supplied power S is not enough to compensate for the power loss O, and the amplitude of oscillations will decay automatically until it reaches A0. Similar considerations show that from A0 the system tends to establish the amplitude A0 as well. The system that demonstrates the given charac- ter of a non-linearity is capable of self-oscillations. In (c) an example of a non-linear function S is given at which oscillations with A = A0 can occur, but they will be unstable. Indeed, setting A > A0 leads to more power entering the system than be- ing lost, and A is pushed to grow further. Setting A < A0 leads to the power spent is not compensated, and consequently to the decrease of A towards zero amplitude, i.e., towards no oscillations. Although this last system can oscillate with a non-zero amplitude A0, it is not a self-sustained system, because such oscillatory regime is not stable: a tiny perturbation will ruin it. If the power of oscillations O is not a linear function of A2, the picture would qualitatively look as in Fig. 2.2. It is qualitatively similar to the one in Fig. 2.1(b), so the same principles apply. Based on these simple considerations, it can be con- cluded that it is an interplay between the non-linear power supply and dissipation that makes self-oscillations possible. Thus, a self-sustained system must be non- linear. Note, that the figures above schematically illustrate the requirements for the simplest periodic self-oscillations to arise, but would not be sufficient to explain the origin of more complex self-oscillations whose amplitude is not constant. How- ever, the fundamental physical principles explained here remain valid for all self- sustained systems, provided the modern developments are taken into account that are discussed in the next paragraph.
- 31. 16 2 General Remarks Fig. 2.2. Powers in the system as functions of the square of oscillations amplitude A2. Dashed line: power O spent on oscillations; solid line: power S supplied into the system. Self-oscillations can occur 2.3.3 Modern Revisions of the Definition of a Self-Sustained System Andronov et al. [12, 14] have defined a self-sustained system as periodic, but nowa- days a family of self-oscillations has expanded considerably to include quasiperi- odic and irregular oscillations, so this requirement is obviously out of date. Also, the original definition required the self-sustained system to be autonomous, which in fact means that the power available from the source should be constant and not depend on time explicitly.8 This definition was revised by Landa [161, 162] in view of modern developments of oscillations theory. She excluded the word “au- tonomous” and has thus allowed the source of energy to change in time. This ad- dition made alone would immediately include oscillations that exist only because of rhythmic external forcing, i.e., forced oscillations which are not self-sustained. However, forced oscillations would have the same or similar time scales as the forc- ing itself. So in order to exclude forced oscillations, Landa adds a requirement that reads “The complete or partial independence of the frequency spectrum of oscil- lations from the spectrum of the energy (power) source” [161]. This means that at least a part of the spectrum of oscillations does not come as a result of the transfor- mation of the spectrum of the source of power, i.e., the frequency components are not harmonics or subharmonics of those of the spectrum of the source. At least a part of the spectrum of oscillations must be defined by the intrinsic properties of the system itself. The relaxation of the condition on the constancy of the power source has an important consequence: it allows one to include into the family of self-sustained oscillations the ones that are induced merely by random perturbations and would not occur without them. In Chap. 9 it will be demonstrated that this classification of noise-induced oscillations is justified, and that they do behave like self-sustained systems in many respects, and in particular can be synchronized. 8 Although the amount of power actually taken from the source at the given time instant does depend on the stage of oscillations, and thus depends on time implicitly.
- 32. 2.4 Duality of the Description of Synchronization 17 2.3.4 Self-Sustained Oscillations and Attractors A distinctive feature of self-oscillating systems is their ability to self-organize. When we launch a process in such a system by, e.g., switching it on, the initial conditions can be chosen at random in a wide range. In general, the time course of a process thus launched can depend on the initial settings quite substantially. However, a self- oscillator is very confident about what it is ought to do, and after some transient (relaxation) time passes by, it arrives at the same regime of oscillations from a large range of initial conditions. In mathematical terms, such regimes are characterized by the attractors in the phase space. Sometimes, certain systems can have a choice of the possible attracting regimes to which they can go, depending on the initial conditions provided, and this is called multistability. Nevertheless, self-sustained systems are generally quite firm in their decisions on how to behave, and are resis- tant to weak attempts to distract them from their course. A mathematical term for this property is robustness. 2.3.5 Synchronization as a Control Tool In various applications it might become necessary to amend the conduct of a self- sustained system either slightly or substantially. One might even want to stop all os- cillations in it. However, this might not be a straightforward and easy task, given the above-mentioned stubbornness of self-oscillators. In this respect, our book shows you the possible ways to control the behavior of self-oscillating systems by means of clever and inexpensive perturbations. But before one is able to choose the best way to tame the particular system, it is necessary to classify it, to learn about the temper and habits of the systems from the given family, and to arm oneself with the full range of the available taming tools. We wish our reader good luck in this exciting journey. 2.4 Duality of the Description of Synchronization Synchronization of oscillations is a phenomenon that was originally discovered by Christian Huygens in 1665 in a mechanical system: two pendulum “grandfather” clocks hanging on the same beam [125]. The interaction between the organ pipes was studied by Rayleigh [243]. The first observations of synchronization in a elec- tronic tube generators were done by Eccles [58, 75] in relation to the problem of creating a precision clock and the transmission of naval signals. Almost at the same time experiments with electric circuits were performed by Appleton [28] and by van der Pol [292, 293] while they were studying the reception of radio signals with electric circuits with triodes. The same authors developed the first theoretical ap- proaches that were able to explain their results to some extent. However, the first non-linear mathematical theory of synchronization which was able to capture the phenomena observed much more accurately, was created in the Soviet Union by Andronov and Witt, also with regard to a very practical problem: stabilization of
- 33. 18 2 General Remarks the frequency of a powerful generator of electromagnetic waves by energy-efficient weak external forcing [13, 299]. In the experiments synchronization was detected by observing Lissajous fig- ures on the screens of oscilloscopes that provided one with information on the phase shifts, amplitudes and frequencies. Thus, synchronization can be naturally described in physical terms like power, frequency or phase. On the other hand, the systems which can demonstrate synchronization, can be described by non-linear mathematical equations. Transitions that occur in coupled systems when their pa- rameters change, can be described in terms of dynamical systems theory including bifurcation theory. We emphasize that the same phenomena can be described using different languages, the language of physics or the language of mathematics. But whatever approach we choose, the underlying phenomena remain the same. In this book we will analyze the phenomenon of synchronization and the associated effects using both languages and making a clear connection between these different levels of description. 2.5 Oscillations Helping Each Other Out A reader who has reached this point in the book might be already thinking: “First, they were talking about my heartbeats, whistles, clocks and bottles, then about some electronic experiments and organ pipes. In between they promised me something exciting to arise out of the coupling of various devices, and also gave a definition of some imaginary self-sustained system. These look like all different things to me, having nothing to do with each other. Even if they are saying that two clocks can be synchronized, so what? How does it help me to understand what happens to organ pipes? And above all, what does it have to do with my heart?” This is a fundamental question which we would be delighted to receive and to answer. We need to make a short excursion into the past. Before the beginning of the 20th century, non-linearity was perceived as an annoying misfortune that could be encountered in this or that physical phenomenon. Every physical problem seemed to contain some non-linearity, but it would be perceived as its own non-linearity specific to the given problem [256], just like you might have suggested above. In the early 1930s Soviet physicist Leonid Mandelstam was the first to recognize the burning need to develop a unique approach to non-linearity and proposed the ideas of non-linear thinking. In addition to that, in 1944 in one of his lectures he made an observation that starting from Kepler laws, most fundamental discoveries made in physics were in fact oscillatory in this or that way. He also observed that oscilla- tions were a key element that was common in all traditional subdivisions of physics: optics, electricity, acoustics, etc. Now we know that oscillations are common in bi- ology, chemistry, geology, finances and social sciences as well, and this list can be continued. His ideas of commonness of oscillations and oscillations’ mutual aid consisted in that there are the same fundamental laws of nature that lie behind os-
- 34. 2.6 Terms of Bifurcations Theory 19 cillations of all kinds. An understanding of the principles behind oscillations in one system would help one to understand oscillations in the other systems. The statement above might not sound immediately obvious, so we continue. Already at the end of the 19th century it was clear that if one considers small os- cillations in acoustics and in electricity, and consistently, from the first principles, derives mathematical equations describing them, the resulting equations will be the same [243]! Moreover, it was shown that the same equations are valid for small oscillations in mechanical systems. Is it a coincidence? Let us go further. Later on, when deriving from the first principles the differ- ential equations underlying the non-small oscillations in the systems of all kinds (chemical, biological, physical) it was noticed that quite often these equations ap- pear equivalent in the sense of topology. The latter means that a change of variables would reduce one set of equations to another, i.e., that there is no real difference between them from the viewpoint of mathematics! At present these ideas are quite well established and might even be occasionally regarded as trivial. But, as Mandelstam once said “It is the triviality of this, which is non-trivial” [256]. Thus, the theory of oscillations serves as a common language that can be spoken by different disciplines. The laws of the theory of oscillations are common between oscillations of the same class, regardless of the nature of the particular system demonstrating them. Coming back to the question in the beginning of this section, we are safe in saying that all seemingly different systems and phenomena that were mentioned here, and a huge lot of those not mentioned, just because there are too many of them, obey the same fundamental principles. If we state that self-oscillations can be synchronized, this means that all self-oscillations can do that, no matter where they are found. In the remainder of Part I of this book the simplest paradigmatic models will be discussed, that describe periodic, chaotic, noisy and noise-induced oscillations. The mathematical results will predict that certain interesting things can happen to them. But then qualitatively the same phenomena can occur even in much more complex systems, provided that their oscillatory properties are equivalent to those described by simple equations. Therefore, when learning about, say, phase locking in van der Pol oscillator, one learns about phase locking in a general periodically self-oscillating system. 2.6 Terms of Bifurcations Theory In the next chapters we use a number of terms that belong to the theory of differ- ential equations, including bifurcation theory. It is not possible to give a detailed introduction into differential equations here, and anyway this is very well done by other authors before us. Just a few useful sources are [65, 101, 156], and we would like to mention separately [2] for those who feel that they need to start from a very basic level. For an excellent historical introduction to dynamical chaos we would mention [3].
- 35. 3 1 : 1 Forced Synchronization of Periodic Oscillations In this chapter we study the simplest case of synchronization: synchronization of unidirectionally coupled periodic oscillators. Another name for this phenomenon is forced synchronization, which reflects the fact that one system influences the other, but does not experience any influence from the other system in return. Another sim- plifying assumption employed here is that the frequency of external stimulus is suf- ficiently close to the frequency of natural, i.e., unforced, oscillations. Provided that the strength and frequency of forcing satisfy certain conditions, a remarkable effect can take place: the system that experiences only weak external perturbation can start to oscillate with the frequency equal to the one of this perturbation. They say that the phenomenon of 1 : 1 phase (frequency) locking, or entrainment, occurs. This is a special case of a more general phenomenon of n : m synchronization that can be observed when the forcing frequency ff is not close to the natural frequency f0 of oscillations in the forced system, but instead is close to a value n m f0. n : m synchro- nization will be considered in Chap. 6. We will start with considering a periodic weakly non-linear oscillator that is forced harmonically. As a particular example we use a famous paradigm for periodic self-sustained oscillations, the van der Pol equation, which has been used to describe a variety of oscillatory phenomena including oscillations of current in electric circuit [292], signal of electrocardiogram [294], dynamics of semiconductor lasers [49],
- 36. 22 3 1 : 1 Forced Synchronization of Periodic Oscillations generation of relativistic magnetrons [168], and the activity of a single neuron [197]. The equation reads ẍ − λ − x2 ẋ + ω2 0x = 0. (3.1) Here, dots over the variables denote derivatives over time t, λ is the non-linearity parameter and also the bifurcation parameter: at λ 0 there are no self-oscillations, and the only stable solution of the system is a stable fixed point at the origin. At λ = 0, Andronov–Hopf bifurcation occurs, as a result of which the fixed point be- comes unstable, and a stable limit cycle is born. At the moment of birth, oscillations on the limit cycle are harmonic, and their frequency is exactly equal to ω0 0, which is also called eigenfrequency. If λ is positive and small, i.e., 0 λ 1, the periodic self-sustained oscillations remain almost harmonic, and their frequency remains approximately equal to the value of ω0. The solution to (3.1) for large t, i.e., after the system has relaxed to the limit cycle from the arbitrarily chosen initial conditions, can be approximately described by x(t) = A cos(ω0t + ϕ0), A = const, ω0 = const, ϕ0 = const, (3.2) where A is the amplitude, ω0 is the frequency, and ϕ0 is the initial phase of oscil- lations. The respective phase portrait on the plane (ẋ, x) and the realization of x(t) are shown in Fig. 3.1 by a black line. This is the case of the so-called weakly non- linear oscillator,1 which can be analyzed analytically by means of the approximate methods of the theory of oscillations. Generally, by a weakly non-linear oscillator we understand a system with a limit cycle, whose control parameters are just above the values corresponding to a super- critical Andronov–Hopf bifurcation.2 Note that when the non-linearity λ in (3.1) is no longer small, the oscillations, although remaining periodic, are no longer close to harmonic, their amplitude grows and the frequency is less than ω0 (Fig. 3.1, grey line). The larger the λ, the slower the oscillations, and the bigger their amplitude is. Now, let us introduce external periodic forcing into the system in its simplest harmonic form as follows: ẍ − λ − x2 ẋ + ω2 0x = B cos(Ωt). (3.3) Here, B and Ω are the strength (amplitude) and frequency of the external forcing, respectively. The solutions of (3.3) at ω0 = 1, fixed small value of B = 0.01 and four different values of Ω close to 1 are illustrated in Fig. 3.2. The external forcing F(t) = B cos(Ωt) (3.4) is shown by black in the fourth column together with the solution x(t). Note that the amplitude B of forcing here is much smaller than the amplitude of x. That is why, in order to allow the reader to compare the details of the behavior of both x and F, in the last column of Fig. 3.2 we show not F, but 10F. 1 Or nearly sinusoidal, as they are called in [12, 14]. 2 There are two forms of Andronov–Hopf bifurcation, a supercritical and a subcritical one. The former is encountered more often, therefore in what follows we will call it simply “Andronov–Hopf bifurcation” for brevity.
- 37. 3 1 : 1 Forced Synchronization of Periodic Oscillations 23 Fig. 3.1. a Phase portraits and b realizations of the autonomous van der Pol oscillator (3.1) at ω0 = 1 and two different values of non-linearity λ: λ = 0.1 (black) and λ = 0.5 (grey) Fig. 3.2. (Color online) Projections of the phase portraits on the planes (ẋ, x) (first column) and (F, x) (second column), Poincaré sections on the plane (F, x) (third column), and realizations x(t) and F(t) = B cos(Ωt) (fourth column) of the forced van der Pol oscillator (3.3) at λ = 0.1, B = 0.01 and different values of Ω: Ω = 0.9, Ω = 0.992, Ω = 1.007, Ω = 1.1 One can see that if the forcing frequency Ω is sufficiently close to the natural frequency ω0 = 1 (Ω = 0.992 and Ω = 1.007), which means that the frequency detuning between the systems is small, the forced oscillations x(t) are periodic. Namely, the phase trajectories tend to the stable limit cycle, and the Poincaré section is a fixed point. Each time F takes its maximal value, x tends to be at the same “stage” of its oscillations. This is phase synchronization of oscillations by external
- 38. 24 3 1 : 1 Forced Synchronization of Periodic Oscillations forcing. Note that synchronized oscillations have constant amplitude, and the values of x at the local maxima are the same from one oscillation to another. However, when the forcing frequency Ω is not close enough to ω0 (Ω = 0.9 and Ω = 1.1), i.e., frequency detuning is not small, an interesting phenomenon occurs: the amplitude of oscillations oscillates itself. This is called beating. The instanta- neous amplitudes, that are roughly half distances between the closest maxima and minima x(t), oscillate periodically with a certain beat frequency. To some extent, this can be visible in the Poincaré section, defined by ẋ = 0, ẍ 0, that shows the maxima against the values of the forcing F taken at the same instants (Fig. 3.2, third column). Later we will consider the beat frequency in more detail and make some theoretical analyses. Perhaps more importantly, when the amplitude of oscillations is not constant the forced oscillations are not synchronous with the forcing: when F takes its maximal values, x can take any value. The projection (F, x) is very informative: it is clearly visible that x and F move independently of each other. In more rigorous terms, the oscillations are quasiperiodic: the phase trajectories lie on the two-dimensional invariant tori whose Poincaré sections are closed curves. This regime corresponds to the absence of synchronization between the system and the forcing. In Fig. 3.2 the 1 : 1 synchronization phenomenon is illustrated numerically. How- ever, for the weakly non-linear oscillator (3.3) considered, synchronization also al- lows for analytical treatment, which will be illustrated in the next sections. 3.1 Phase of Quasiharmonic Oscillations Here, we need to introduce an important idea closely associated with the phenom- enon of synchronization—the idea of phase. When discussing the synchronization illustrated in Fig. 3.2, we mentioned the “stage” of oscillations, which is the current position of the system inside the given cycle of oscillations, e.g., the beginning, the first quarter, the middle, the end, etc. We need a quantity characterizing the “stage” of oscillations at any given time moment t: call it phase ψ(t). For a purely har- monic function of time like in (3.2) the phase can be introduced uniquely as an argument of the cosine or sine, which in the given case will be ψ(t) = ω0t + ϕ0. In Fig. 3.3(a) a harmonic function cos(1.005t + 0.5) is shown by a solid line that represents harmonic forcing. The period of the cosine function is 2π, so if we start to observe the cosine at some time moment t, the onset of the nth full os- cillation cycle (n = 1, 2, . . .) can be characterized by the values of ψ(t) equal to ϕ0 +2π(n−1), and the end of it by ϕ0 +2πn. Thus, the first oscillation cycle will be within ψ(t) ∈ [ϕ0; ϕ0 + 2π], and in terms of time inside the interval t ∈ [0; 2π/ω]. For harmonic oscillations, phase is a linear function of time. In Fig. 3.3(c) the phase ψ(t) = (1.005t + 0.5) of the signal in (a) is shown with filled circles. In (a) filled circles indicate the values of the signal at the instants t = 2πn/1.005, i.e., when phase ψ(t) changes by 2π. In this chapter we will deal with oscillations in a forced weakly non-linear sys- tem. Such oscillations, generally not being harmonic, can be viewed as almost har-
- 39. 3.1 Phase of Quasiharmonic Oscillations 25 Fig. 3.3. Illustration of phase of quasiharmonic oscillations. a Harmonic signal cos(1.005t + 0.5) that represents forcing; b quasiharmonic signal x(t) that represents response of the forced system; c phases of a harmonic forcing (filled circles) and quasiharmonic response (solid line); d phase difference ϕ(t) between the response and the forcing that oscillates slightly around some constant and is thus an evidence of 1 : 1 phase synchronization monic, or “quasiharmonic.” This term means that the oscillations can be described as cosine (or sine) whose argument is not a linear function of time, but is close to being linear, and whose amplitude is not a constant, but changes either slowly, or slightly. For quasiharmonic oscillations phase can be also introduced as an argument of cosine. An example of quasiharmonic oscillations is given in Fig. 3.3(b), and its phase in (c) by solid line. It has to be mentioned that purely harmonic or quasiharmonic oscillations are rare in real life. Unfortunately, even if oscillations are periodic, but non-harmonic, there is no unique way to introduce a phase. In more detail, the problem of intro- duction of a phase for non-periodic oscillations of complex shape, including chaotic ones, will be discussed in Sect. 8.3. Because in this chapter we are not going to con- sider any other oscillations besides the weakly non-linear ones, we do not need to be bothered with the more difficult cases right now. Note that phase itself can serve as a useful instrument for describing the oscilla- tions. However, in relation to the synchronization problem, phase represents a con- venient tool for detection whether two oscillations are synchronized or not. Namely, one can introduce phases for the two oscillations and consider their difference ϕ that is usually referred to as “phase difference.” If the phase difference happens to be a constant or to oscillate slightly around a constant3 this would usually imply that two oscillations are 1 : 1 synchronized. An illustration of this is given in Fig. 3.3(d) that shows the phase difference between the forcing in (a) and the response in (b). If the phase difference grows or decreases monotonously in time, there is no 1 : 1 synchronization. 3 If phase difference oscillates around some constant, it does not necessarily mean synchro- nization. For more detail, see the description of Fig. 3.9.
- 40. 26 3 1 : 1 Forced Synchronization of Periodic Oscillations 3.2 Derivation of Truncated Equations for Phase Difference and Amplitude It should be noted that the exact oscillating solution of (3.3) for the arbitrary values of parameters λ, B and Ω cannot be found analytically with the mathematical tools available so far. However, for a certain range of values of these parameters, one can analytically find the approximate solutions that would describe the unknown exact solutions with a certain degree of accuracy. This would be quite sufficient from the practical viewpoint, since whatever is measured in a real experiment is anyway measured with a certain error. Hence, approximate theory can be good enough when one compares it with an experiment. The idea of calculations similar to the ones presented here first occurred to Andronov and Witt in the 1920s [13, 299]. The significance of their results is that they were one of the first to successfully analyze the non-linear systems by the approximate methods, and to provide an accurate explanation of the earlier experimental observations of synchronization in electric circuits. The analysis of equations of the form similar to that of (3.3), i.e., of non- linear, dissipative ordinary differential equations of the second order with weak non- linearity and with periodic excitation, was also done by Cartwright [61, 62], Gillies [88], Holmes and Rand [117], Arrowsmith [36]. An introduction into this analysis was made in [179] and [110] with more references. In what follows, we will restrict our analysis to the small values of λ, for which without the forcing (B = 0) the solution is almost harmonic (3.2). The addition of forcing (B = 0) will obviously change the solution. However, let us assume that the forcing is not too strong, i.e., B is not large as compared to the amplitude A0 of unperturbed self-oscillations, and the forcing frequency Ω is only slightly different from ω0. Then the solution of (3.3) can be approximately described as a quasiharmonic oscillation, i.e., oscillation in the form of (3.2), whose amplitude A and the argument ψ (phase) of the cosine are perturbed by the forcing. Since the system, (3.3), under study is close to being linear with small λ, it is natural to suppose that its response to an external forcing at the frequency Ω contains the frequency component Ω. We will thus be looking for a solution in the form of a quasiharmonic function of time, namely, x(t) = A(t) cos(Ωt + ϕ(t)). (3.5) Here, A(t) is the envelope of the oscillations x(t) illustrated by Fig. 3.2 (fourth column). A does not change in time when synchronization occurs, and oscillates slowly when beating starts. We assume that both A(t) and ϕ(t) are slow functions of time compared to the function cos(Ωt). Mathematically, this condition can be written as Ȧ(t) ΩA(t), |ϕ̇(t)| Ω. (3.6) The full phase ψ(t) of the forced oscillations is ψ(t) = Ωt + ϕ(t), (3.7)
- 41. 3.2 Derivation of Truncated Equations for Phase Difference and Amplitude 27 while the phase of forcing ψf (t) is ψf (t) = Ωt. Hence, ϕ(t) is the phase difference between the forcing and the forced oscillations. When ϕ(t) is a constant, oscillations x(t) in the system are 1 : 1 synchronized by the external forcing and are harmonic with frequency Ω. When ϕ(t) changes in time, there is no 1 : 1 synchronization. Thus, in order to reveal the synchronization conditions, we need to formulate the explicit equations describing the evolution of ϕ and A in time. Synchronization will mean that there is/are stable fixed point/s in these equations, so we will have to find the conditions for these points to exist and to be stable. To derive the equations for ϕ and A, one can use the method of averaging, also known as the Krylov–Bogoliubov4 method [150]. In the following, for brevity we will omit the brackets “(t)” denoting the explicit dependence on time of A and ϕ. If we calculate the time derivative of x(t) rigorously, we obtain ẋ = Ȧ cos(Ωt + ϕ) − AΩ sin(Ωt + ϕ) − Aϕ̇ sin(Ωt + ϕ). (3.8) By representing the solution in the form of (3.5), instead of one independent phase variable x(t) we introduce two phase variables: A(t) and ϕ(t). Thus, an ambiguity is introduced into the system. In order to remove the introduced ambiguity, we have to specify an additional condition that A(t) and ϕ(t) should satisfy. It is convenient to set such a condition that the derivative of x(t) is a simple expression in the form ẋ = −AΩ sin(Ωt + ϕ), (3.9) which would immediately imply Ȧ cos(Ωt + ϕ) − Aϕ̇ sin(Ωt + ϕ) = 0. (3.10) Next, we need to find ẍ. The calculations can be continued using the expressions above that contain sines and cosines, but it is usually more convenient to operate with exponential functions. Thus, we want to express all trigonometric functions in x(t), ẋ and ẍ in terms of exponents of complex arguments. We start from reformu- lating the solution, (3.5), x = A cos(Ωt + ϕ) = A ei(Ωt+ϕ) + e−i(Ωt+ϕ) 2 = eiΩt Aeiϕ + e−iΩt Ae−iϕ 2 . Let us introduce a complex function of time a, such that a = Aeiϕ , a∗ = Ae−iϕ , (3.11) where the asterisk denotes the complex conjugate. Then x(t) can be represented through a as 4 Bogoliubov is also sometimes spelled as Bogolyubov or Bogolioubov in literature.
- 42. 28 3 1 : 1 Forced Synchronization of Periodic Oscillations x = 1 2 aeiΩt + a∗ e−iΩt . (3.12) We can call a a complex amplitude of oscillations. The condition (3.10) can be rewritten as Ȧ ei(Ωt+ϕ) + e−i(Ωt+ϕ) 2 − Aϕ̇ ei(Ωt+ϕ) − e−i(Ωt+ϕ) 2i = eiΩt Ȧeiϕ + e−iΩt Ȧe−iϕ 2 − eiΩt Aϕ̇eiϕ − e−iΩt Aϕ̇e−iϕ 2i = 1 2 eiΩt Ȧeiϕ + iAϕ̇eiϕ + 1 2 e−iΩt Ȧe−iϕ − iAϕ̇e−iϕ = 0. With the account of the following: ȧ = Ȧeiϕ + Aiϕ̇eiϕ and ȧ∗ = Ȧe−iϕ − Aiϕ̇e−iϕ , the condition (3.10) turns into ȧeiΩt + ȧ∗ e−iΩt = 0. (3.13) Now, consider ẋ and rewrite (3.9) as ẋ = −AΩ ei(Ωt+ϕ) − e−i(Ωt+ϕ) 2i = iΩ 2 aeiΩt − a∗ e−iΩt . (3.14) Consider ẍ as a derivative of (3.14) ẍ = iΩ 2 ȧeiΩt + aiΩeiΩt − ȧ∗ e−iΩt + a∗ iΩe−iΩt = iΩ 2 ȧeiΩt − Ω2 2 aeiΩt − iΩ 2 ȧ∗ e−iΩt − Ω2 2 a∗ e−iΩt . Add and subtract iΩ 2 ȧeiΩt and regroup terms ẍ = iΩȧeiΩt − iΩ 2 ȧeiΩt − Ω2 2 aeiΩt − iΩ 2 ȧ∗ e−iΩt − Ω2 2 a∗ e−iΩt . The sum of the second and the fourth terms satisfies the condition (3.13) and is equal to zero. Hence ẍ = iΩȧeiΩt − Ω2 1 2 aeiΩt + a∗ e−iΩt . (3.15) Substitute x, ẋ and ẍ ((3.12), (3.14), (3.15), respectively) into (3.3) iΩȧeiΩt − Ω2 2 aeiΩt + a∗ e−iΩt − λ − 1 4 aeiΩt + a∗ e−iΩt 2 iΩ 2 aeiΩt − a∗ e−iΩt + ω2 0 2 aeiΩt + a∗ e−iΩt = B eiΩt + e−iΩt 2 .
- 43. 3.2 Derivation of Truncated Equations for Phase Difference and Amplitude 29 Regroup terms iΩȧeiΩt + (ω2 0 − Ω2) 2 aeiΩt + a∗ e−iΩt − λ iΩ 2 aeiΩt + λ iΩ 2 a∗ e−iΩt + 1 4 a2 ei2Ωt + a∗2 e−i2Ωt + 2aa∗ iΩ 2 aeiΩt − a∗ e−iΩt = B eiΩt + e−iΩt 2 . Open the brackets iΩȧeiΩt + (ω2 0 − Ω2) 2 aeiΩt + a∗ e−iΩt − λ iΩ 2 aeiΩt + λ iΩ 2 a∗ e−iΩt + iΩ 8 a3 ei3Ωt − iΩ 8 a2 a∗ eiΩt + iΩ 8 aa∗2 e−iΩt − iΩ 8 a∗3 e−i3Ωt + iΩ 4 a2 a∗ eiΩt − iΩ 4 aa∗2 e−iΩt = B eiΩt + e−iΩt 2 . (3.16) Collect similar terms and multiply the whole equation by e−iΩt /(iΩ), ȧ + (ω2 0 − Ω2) 2iΩ a + a∗ e−i2Ωt − λ 2 a + λ 2 a∗ e−i2Ωt + 1 8 a3 ei2Ωt + 1 8 a2 a∗ − 1 8 aa∗2 e−i2Ωt − 1 8 a∗3 e−i4Ωt = B 2iΩ 1 + e−i2Ωt . (3.17) We remind you, that the aim of our calculations is to write down the equations describing the evolution in time of the complex amplitude a(t), and then to solve them. Knowing a(t), we will know A and ϕ, and thus we will know the approximate solution x(t) of van der Pol equation (3.3). However, (3.17) is not simpler than the original (3.3), and it is not easier to find the amplitude a from it than it was to find x from (3.3). To simplify the problem we can make more use of the fact of slowness of A and ϕ and exploit the method of averages by Krylov and Bogoliubov [150]. Note that a, ȧ and a∗ are slow functions of time as compared to the functions e±nΩt , n being an integer number. This means that they almost do not change during one period of fast oscillations with the frequency Ω. If we average the whole equation over one period T = 2π/Ω of fast oscillations, we can get rid of the fast terms, and only the slow terms will remain in the equation. The time average ¯ f T of a smooth function f (t) over the time interval T is defined as follows: ¯ f T = 1 T t0+T t0 f (t) dt. (3.18)
- 44. 30 3 1 : 1 Forced Synchronization of Periodic Oscillations Consider terms in (3.17) containing e−i2Ωt . The time average of the second such term is equal to 1 T t0+T t0 λ 2 a∗ e−i2Ωt dt ≈ λ 2 a∗ Ω 2π t0+2π/Ω t0 e−i2Ωt dt = λ 2 a∗ Ω 2π 1 −2iΩ e−i2Ωt t0+2π/Ω t0 = λ 2 a∗ Ω 2π 1 −2iΩ cos(2Ωt) t0+2π/Ω t0 =0 − i sin(2Ωt) t0+2π/Ω t0 =0 = 0. By analogy, it is easy to show that the average values of terms containing ei2Ωt and e−i4Ωt are equal to zero as well. Thus, we obtain the time-averaged equations which, with account of a2a∗ = a(aa∗) = a|a|2, read ȧ + (ω2 0 − Ω2) 2iΩ a − λ 2 a + 1 8 a|a|2 = −i B 2Ω . (3.19) Recall that a = Aeiϕ and substitute it into (3.19) Ȧeiϕ + Aiϕ̇eiϕ − i (ω2 0 − Ω2) 2Ω Aeiϕ − λ 2 Aeiϕ + 1 8 A3 eiϕ = −i B 2Ω . Multiply everything by e−iϕ Ȧ + Aiϕ̇ − i (ω2 0 − Ω2) 2Ω A − λ 2 A + 1 8 A3 = −i B 2Ω e−iϕ . Introduce the frequency detuning Δ between the unperturbed system and the forcing Δ = (ω2 0 − Ω2) 2Ω ≈ (ω0 − Ω), (3.20) the latter approximation being valid when the forcing frequency Ω is close to the natural frequency of unperturbed oscillations ω0 (Ω ≈ ω0). Represent e−iϕ through cos ϕ and sin ϕ Ȧ + iAϕ̇ − iAΔ − λ 2 A + 1 8 A3 = −i B 2Ω e−iϕ = −i B 2Ω (cos ϕ − i sin ϕ). Separate the real and imaginary parts of the equation Ȧ − λ 2 A + 1 8 A3 = − B 2Ω sin ϕ, Aϕ̇ − AΔ = − B 2Ω cos ϕ.
- 45. 3.3 Amplitude of Unperturbed Oscillations at Small Non-linearity 31 Finally, we obtain Ȧ = λ 2 A − 1 8 A3 − B 2Ω sin ϕ, (3.21) ϕ̇ = Δ − B 2AΩ cos ϕ. (3.22) These are the famous truncated equations for the amplitude A of forced oscillations and for the phase difference ϕ between the latter and the external forcing. These equations have a fundamental importance in the theory of synchronization. Their significance is due to the fact that the analysis of a van der Pol equation (3.3) that is non-autonomous, i.e., depends on time explicitly, is reduced to the analysis of the autonomous system of equations (3.21)–(3.22). In terms of bifurcation theory, instead of analyzing periodic orbits of (3.3), we can analyze the fixed points in (3.21)–(3.22), which is obviously much easier. The fixed points of (3.21)–(3.22) mean that the phase difference between the system and external forcing does not change in time (ϕ = const), i.e., the external forcing has synchronized the system, and the oscillations are periodic with constant amplitude A and the frequency of external forcing Ω. Thus, finding of the conditions when these fixed points are stable will mean finding the conditions at which 1 : 1 forced synchronization occurs. 3.3 Amplitude of Unperturbed Oscillations at Small Non-linearity It is clearly seen that both truncated equations (3.21)–(3.22) are non-linear, A and ϕ influencing each other in the presence of forcing (B = 0). If there is no forcing (B = 0), one can estimate the stationary amplitude of natural self-oscillations, i.e., the amplitude that the oscillations will have after the sufficiently long relaxation time will pass. In order to do this, in (3.21) one should set Ȧ = 0 and solve the algebraic equation fA(A) = λ 2 A − 1 8 A3 = 0, A0 = 0, (3.23) A0 = 2 √ λ. Solution A0 = 0 corresponds to the absence of oscillations, i.e., to the fixed point. Strictly speaking, there are two roots corresponding to the non-zero solution, but only the positive one makes sense, since the amplitude is supposed to be a positive value by definition. The stability of the fixed points is determined by the sign of ∂fA(A)/∂A: if it is negative (positive), the point is stable (unstable): ∂fA(A) ∂A A=0 = λ 2 − 3 8 A2 A=0 = λ 2 0, ∂fA(A) ∂A A=2 √ λ = λ 2 − 3 8 A2 A=2 √ λ = λ 2 − 3 8 4λ = −λ 0.
- 46. 32 3 1 : 1 Forced Synchronization of Periodic Oscillations Thus, the non-oscillatory solution is unstable, and the oscillatory one is stable. In what follows let us denote the amplitude of natural (unperturbed) self-oscillations as A0. Hence, A0 is proportional to the square root of the non-linearity parameter λ while the latter remains small. 3.4 Analysis of Truncated Equations for Weak Forcing Consider the non-zero forcing. The analysis of these equations for arbitrary values of B and Ω is difficult. However, a few special cases can be considered that allow for approximate analytical solutions. In the simplest case when the strength B of forcing can be regarded as very small B εA0, (3.24) the amplitude of the perturbed oscillations is not very different from A0. In the equation for ϕ we can set A = A0 as in (3.23), and then it becomes independent of A ϕ̇ = Δ − B 4 √ λΩ cos ϕ = fϕ(ϕ). (3.25) The fixed points of this equation that correspond to ϕ̇ = 0 can be found by solving a non-linear algebraic equation cos ϕ = 4 √ λΩΔ B , (3.26) which is illustrated in Fig. 3.4. One can see that cos ϕ can intersect the horizontal line twice, thus there can be two solutions ϕ1 = cos−1 4 √ λΩΔ B , ϕ2 = 2π − cos−1 4 √ λΩΔ B , which exist provided that 4 √ λΩ|Δ| ≤ B. (3.27) Their stability is determined by the sign of ∂fϕ(ϕ)/∂ϕ in (3.25). Namely, ∂fϕ(ϕ) ∂ϕ ϕ1 = B 4 √ λΩ sin cos−1 4 √ λΩΔ B = B 2A0Ω 1 − cos2 cos−1 4 √ λΩΔ B = B 4 √ λΩ 1 − 4 √ λΩΔ B 2 ≥ 0
- 47. 3.4 Analysis of Truncated Equations for Weak Forcing 33 Fig. 3.4. Graphical illustration of the solution of the non-linear equation (3.26) and ∂fϕ(ϕ) ∂ϕ ϕ2 = B 4 √ λΩ sin 2π − cos−1 4 √ λΩΔ B = B 4 √ λΩ sin 2π × cos cos−1 4 √ λΩΔ B − cos 2π × sin cos−1 4 √ λΩΔ B = − B 4 √ λΩ 1 − cos2 cos−1 4 √ λΩΔ B = − B 4 √ λΩ 1 − 4 √ λΩΔ B 2 ≤ 0, which means that the fixed point ϕ2 is stable and ϕ1 is unstable. When the strict equality in (3.27) is satisfied, two fixed points merge, and when (3.27) is no longer valid, the pair of fixed points disappear via saddle-node bifurcation. This means that there is no longer a constant phase difference between the forcing and the response in (3.3). Hence, the equation B = 4 √ λΩ|Δ| (3.28) describes the borderline of 1 : 1 synchronization region at very small strengths of forcing B. At λ = 0.1 and ω0 = 1, and with the approximation in (3.20) for Δ, the syn- chronization region defined by (3.28) is outlined by the shaded area in Fig. 3.5 on the plane of forcing parameters (Ω, B). We see that it has the characteristic shape of a tongue with a tip at Ω ≈ ω0 = 1. The solid lines show the numerically esti- mated lines of saddle-node bifurcations of a stable and a saddle periodic orbits of the original non-autonomous equation (3.3) for the same parameters. We see that the approximation for the synchronization region border given by (3.28) is quite accurate for a significant range of B.
- 48. 34 3 1 : 1 Forced Synchronization of Periodic Oscillations Fig. 3.5. (Color online) 1 : 1 synchronization tongue for the forced van der Pol system (3.3) on the plane of forcing parameters Ω and B, at λ = 0.1, ω0 = 1. Lines correspond to bifurcations of periodic solutions: solid lines mark saddle-node bifurcations, dashed lines mark torus birth (Neimark–Sacker) bifurcations, both obtained numerically from the direct analysis of (3.3). Shaded area shows analytical prediction of the locking region according to (3.28). Insets to the right show the connections between the above types of lines in more detail (compare with insets in Fig. 3.7) 3.5 Derivation of Truncated Equations in Descartes Coordinates If the amplitude B of forcing is not vanishingly small, the truncated equations (3.21)–(3.22) for the amplitude A and phase ϕ of the complex amplitude a that mod- ulates periodic oscillations according to (3.12) can have up to three fixed points. To reveal the borderlines of the synchronization region, one could find the fixed points, analyze their stability and find the lines in the parameter plane on which bifurcations occur. However, although the equations look quite compact, their analysis is quite involved. It appears that if the same equations are rewritten in Descartes coordinates instead of the polar ones, their analysis becomes less cumbersome. In this section we show how to obtain the Descartes form of the truncated equations. At the arbitrary values of B, A can no longer be regarded as a constant ap- proximately equal to the amplitude of the unperturbed oscillations A0, and the two equations cannot be separated. Perform the following variable substitution: ū(t) = A(t) cos ϕ(t), v̄(t) = A(t) sin ϕ(t). (3.29) The time derivatives of the new variables can be expressed through A and ϕ as ˙ ū = Ȧ cos ϕ − Aϕ̇ sin ϕ, ˙ v̄ = Ȧ sin ϕ + Aϕ̇ cos ϕ, and further with account of (3.21)–(3.22) as
- 49. 3.5 Derivation of Truncated Equations in Descartes Coordinates 35 ˙ ū = cos ϕ λ 2 A − 1 8 A3 − B 2Ω sin ϕ − A sin ϕ Δ − B 2AΩ cos ϕ , (3.30) ˙ v̄ = sin ϕ λ 2 A − 1 8 A3 − B 2Ω sin ϕ + A cos ϕ Δ − B 2AΩ cos ϕ . Note that A = ū2 + v̄2, cos ϕ = ū √ ū2 + v̄2 , sin ϕ = v̄ √ ū2 + v̄2 , and substitute this into (3.30), ˙ ū = ū √ ū2 + v̄2 λ 2 ū2 + v̄2 − 1 8 ū2 + v̄2 3/2 − B 2Ω v̄ √ ū2 + v̄2 − ū2 + v̄2 v̄ √ ū2 + v̄2 Δ − B 2Ω √ ū2 + v̄2 ū √ ū2 + v̄2 = ū λ 2 − ū 8 ū2 + v̄2 − Būv̄ 2Ω(ū2 + v̄2) − v̄Δ + Būv̄ 2Ω(ū2 + v̄2) , ˙ v̄ = v̄ √ ū2 + v̄2 λ 2 ū2 + v̄2 − 1 8 ū2 + v̄2 3/2 − B 2Ω v̄ √ ū2 + v̄2 + ū2 + v̄2 ū √ ū2 + v̄2 Δ − B 2Ω √ ū2 + v̄2 ū √ ū2 + v̄2 = v̄ λ 2 − v̄ 8 ū2 + v̄2 − Bv̄2 2Ω(ū2 + v̄2) + ūΔ − Bū2 2Ω(ū2 + v̄2) . The third and the fifth terms of the right-hand part of the equation for ˙ ū cancel each other. Collect similar terms and rewrite the set of equations for ˙ ū and ˙ v̄, ˙ ū = ū 2 λ − (ū2 + v̄2) 4 − v̄Δ, (3.31) ˙ v̄ = v̄ 2 λ − (ū2 + v̄2) 4 + ūΔ − B 2Ω . (3.32) Note that (3.31)–(3.32) are completely equivalent to (3.21)–(3.22) and describe ex- actly the same kind of behavior, only in different coordinates. Typical phase portraits of these two types of equations are shown in Fig. 3.6 for parameter values λ = 0.1, Ω = 1.005, B = 0.01 (for reference, see the bifurcation diagram in Fig. 3.5): in po- lar coordinates in Fig. 3.6(a), and in Descartes coordinates in Fig. 3.6(b). The phase difference ϕ is shown by the modulus of 2π. At these parameters phase locking takes place, and there are two fixed points in the truncated equations: one saddle (empty circle) and one stable (filled circle). Note that the closed curve formed by the manifolds of the saddle point in Fig. 3.6(b) is almost a perfect circle. One might think that for any value of ϕ the amplitude A should take the same values, but as seen from Fig. 3.6(a), this is not the case. The
- 50. 36 3 1 : 1 Forced Synchronization of Periodic Oscillations Fig. 3.6. Phase portraits of the truncated equations for the amplitude A and phase difference ϕ corresponding to the forced van der Pol oscillator: a in polar coordinates (A, ϕ) from (3.21)–(3.22); b in Descartes coordinates (ū, v̄) from (3.31)–(3.32). Parameters are λ = 0.1, Ω = 1.005, B = 0.01 and correspond to the phase locking. Filled (empty) circles show stable (saddle) point. Solid lines show the unstable manifolds of the saddle points reason is that the amplitude A is the distance between the phase point and the origin in Fig. 3.6(b). With this, the center of this circle is not at the origin, so at different values of ϕ, the distance between the origin and the phase point is different. It is not convenient to analyze (3.31)–(3.32) in their present form, since they contain too many control parameters. For the further analysis, let us try to make them look simpler. Denote u = ū 2 √ λ , v = v̄ 2 √ λ , (3.33) substitute into (3.31)–(3.32) and divide the result by 2 √ λ du dt = λ u 2 1 − u2 + v2 − vΔ, (3.34) dv dt = λ v 2 1 − u2 + v2 + uΔ − B 4 √ λΩ . Introduce new independent argument τ, τ = λt 2 , and divide both equations by λ/2, du dτ = u 1 − u2 + v2 − v 2Δ λ , dv dτ = v 1 − u2 + v2 + u 2Δ λ − B 2Ωλ √ λ .
- 51. 3.6 Analysis of Truncated Equations in Descartes Coordinates 37 Denote δ = 2Δ λ , F = B 2Ωλ √ λ (3.35) and rewrite the last equations as du dτ = u 1 − u2 + v2 − δv = f (u, v), (3.36) dv dτ = v 1 − u2 + v2 + δu − F = g(u, v). (3.37) These are the equations for a non-linear system that is potentially able to demon- strate self-sustained periodic oscillations. 3.6 Analysis of Truncated Equations in Descartes Coordinates We will now analyze the stability of the fixed points of (3.36)–(3.37) without making any simplifying assumptions on the values of parameters δ and F. The fixed points are defined by u̇ = f (u, v) = 0, v̇ = g(u, v) = 0. (3.38) From (3.36) we obtain δv = u 1 − u2 + v2 , (3.39) and from (3.37) 1 − u2 + v2 = F − δu v . (3.40) Substitute (3.40) into (3.39) v2 = u(F − δu) δ . (3.41) Substitute v2 from (3.41) into (3.40) (F − δu) = v 1 − u2 − Fu δ + u2 = v 1 − Fu δ , (3.42) v = δ F − δu δ − Fu . (3.43) Take a square of the last expression to obtain v2 = δ2 (F − δu)2 (δ − Fu)2 . (3.44) Equating (3.44) and (3.41) leads to the following equation that only includes the u-variable: δ2 (F − δu)2 (δ − Fu)2 = u(F − δu) δ . (3.45)
- 52. 38 3 1 : 1 Forced Synchronization of Periodic Oscillations (F − δu) = 0 could be a root of (3.45). But due to (3.43), it would lead to v = 0, which is not a root of (3.38). Thus, (F − δu) = 0, and we can safely divide by it both parts of (3.45) δ3(F − δu) (δ − Fu)2 = u. Simple transformation leads to the cubic equation for u F2 u3 − 2Fδu2 + δ2 + δ4 u − δ3 F = 0. (3.46) We need to solve this equation. Denote ã = F2 , b̃ = −2Fδ, c̃ = δ2 + δ4 , d̃ = −δ3 F. (3.47) The first step in solving a cubic equation is obtaining a “depressed cubic equation,” i.e., equation without a quadratic term, by making the following variable substitu- tion: u = u∗ − b̃ 3ã . (3.48) Substitute u in the above form into (3.46) and obtain u3 ∗ + c̃ ã − b̃2 3ã2 u∗ + 2b̃3 27ã3 − b̃c̃ 3ã2 + d̃ ã = 0. (3.49) Denote C = c̃ ã − b̃2 3ã2 , D = 2b̃3 27ã3 − b̃c̃ 3ã2 + d̃ ã , (3.50) so that (3.49) becomes u3 ∗ + Cu∗ + D = 0. (3.51) Express C and D through δ and F using (3.47) C = δ2 + δ4 F2 − 4F2δ2 3F4 = 3δ4 − δ2 3F2 , (3.52) D = 2 × (−2Fδ)3 27F6 − (−2Fδ) × (δ2 + δ4) 3F4 + (−δ3F) F2 = δ3 27F3 18δ2 − 27F2 + 2 . (3.53) A cubic equation (3.51) has either three real roots, or one real and two complex- conjugate roots. With this, either all three, or two of three, roots can coincide, but there is always at least one real root u∗1. If we find this real root, we can divide (3.51) by (u∗ − u∗1), obtain a quadratic equation and then find the two remaining roots. u∗1 can be found from (3.51) by substitution u∗1 = s − t (3.54)
- 53. 3.6 Analysis of Truncated Equations in Descartes Coordinates 39 with s and t such that 3st = C, (3.55) s3 − t3 = −D. (3.56) If we express s through t from the first equation and substitute into the second one, we obtain an equation of the sixth order with respect to t t6 − Dt3 − C3 27 = 0. Denoting z = t3, we obtain a quadratic equation with respect to z z2 − Dz − C3 27 = 0. (3.57) The solution of this equation is z = 1 2 D ± D2 + 4C3 27 = 1 2 (D ± √ R), (3.58) where R is R = D2 + 4C3 27 . (3.59) Because of the sign “±,” there are two solutions for z = t3, but we can take either one of them: whatever sign we choose before √ R, the final solution u∗1 will not depend on this choice. The properties of the cubic equations are such that if R 0, there are three real roots in (3.46); if R 0, there is only one real root in (3.46), and the other two are complex and thus not “physical.” When R = 0, two of the three real roots u∗1, u∗2 and u∗3 coincide. We remind you that u∗j , j = 1, 2, 3 are the shifted and rescaled components of the amplitudes Aj modulating periodic oscillations in the forced van der Pol equation (see (3.29)). If at certain values of the normalized detuning δ and normalized forcing strength F, two of these amplitudes coincide, this means that a saddle-node bifurcation occurs to the fixed points of the system (3.36)–(3.37) and to the periodic orbits of the original forced van der Pol system (3.3). Thus, the equation R = D2 + 4C3 27 = 0 (3.60) is the condition of a saddle-node bifurcation. Let us reveal the equation of the line of saddle-node bifurcation on the plane of parameters (δ, F) by substituting into (3.60) the expressions for C and D from (3.52) and (3.53) R = δ6 272F6 18δ2 − 27F2 + 2 2 + 4 27 3δ4 − δ2 3F2 3 (3.61) = 4δ6 F6 F4 4 − F2 27 1 + 9δ2 + δ2 27 δ2 + 1 2 . (3.62)
- 54. 40 3 1 : 1 Forced Synchronization of Periodic Oscillations Fig. 3.7. 1 : 1 forced synchronization tongue on the plane (δ, F) of the parameters of the truncated equations in the Descartes coordinates (3.36)–(3.37). Saddle-node (solid line) and Andronov–Hopf (dashed line) bifurcation lines of the fixed points are calculated exactly and are described by (3.64) and (3.72), respectively. Insets to the right show the connections between two types of bifurcation lines in more detail: the upper one is accurate, and the lower one is schematic which is included in order to emphasize the configuration of the curves Thus, the curve defined by the condition F4 4 − F2 27 1 + 9δ2 + δ2 27 δ2 + 1 2 = 0 (3.63) is the line of saddle-node bifurcation. We can plot this line denoted by FSN by noting that the condition above defines a quadratic equation with respect to F2. We can solve this equation and find two branches FSN 1 (δ) and FSN 2 (δ) of the saddle-node line FSN 1,2 (δ) = √ 2 (1 + 9δ2) 27 ± (1 + 9δ2)2 272 − δ2 27 δ2 + 1 2 1/2 , (3.64) taking the positive values of FSN 1,2 . FSN is symmetric with respect to δ = 0, and is shown in Fig. 3.7 by a solid line. One can see that FSN outlines a closed region in the (δ, F) plane. It can be easily shown, e.g., by trying just one point inside this area, that R is negative there. Thus there are three real roots of (3.46), implying three fixed points in (3.36)–(3.37), and hence there are three possible values of fixed amplitude of periodic oscillations of the original forced van der Pol oscillator (3.3). The analysis of the eigenvalues of the fixed points of (3.36)–(3.37) reveals that only one point is stable. Outside the region bounded by the saddle-node bifurcation line, R is positive, hence there is only one fixed point in (3.36)–(3.37). There is also a special point in the (δ, F) parameter plane, corresponding to C = D = 0, at which all three roots of (3.51) coincide and equal to zero. This means that three fixed points of (3.36)–(3.37), and the three respective periodic orbits of (3.3), merge. From (3.52) it follows that
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- 56. The hopper is securely fastened on top of the baseboard and over the cylinder. The concave is slipped into place and held with wedges or by driving two nails in just far enough to fasten it temporarily. The concave can be adjusted for grinding the different vegetable products, or replaced at any time with a new one. The ends of the base are supported on boxes, or legs may be provided if desired. When grinding cabbage, cut the heads into quarters and remove the hearts. Press the cabbage on the cylinder and turn the crank. Fine bits of cabbage, suitable for sauerkraut, will be the result.—Contributed by J. G. Allshouse, Avonmore, Pa.
- 57. Opening for Air at the Top of a Shade Procure an extra long shade and cut two openings in the end to be used at the top. The openings may be cut square or ornamental as desired, leaving a strip at each side and one in the center. These strips are reinforced by gluing on some of the same material as the shade or pieces of tape. When the Shade is Pulled Down the Openings Coincide with the Opening over the Upper Sash A shade made in this manner permits the air to enter the room unhindered when the top sash is lowered and at the same time obstructs the view of passers-by.—Contributed by Warren E. Crane, Cleveland, O.
- 58. Hose Attachment for Watering Window Plants The window garden of the house has its watering difficulties which one owner overcame in a neat and handy manner. A hose on a weighted reel was attached to the joists in the basement under the floor near the window flower pots. The weight on the reel kept the hose wound on it and the nozzle end which projects through the floor is large enough to hold it from passing through the hole bored for the hose. A long stem valve was provided with the wheel attached above the floor for turning the water on and off. The Hose is Automatically Run on a Reel by a Weight beneath the Floor When the plants need a shower all that is necessary is to draw the hose nozzle up and turn on the water. The hole for the hose and the valve wheel can be located close to the wall under the flower tray where they will scarcely be seen.
- 59. Removing Paint from Glass Paint may be easily cleaned from glass by using a 50-per-cent solution of acetic acid. The acid should be heated and applied with a cloth. The hot acid will not hurt the hands or fabrics, nor the glass, but should be kept from children who might drink of it. The solution is made of commercial acetic acid and heated by adding hot water. The acid is inexpensive and can be purchased at any local drug store.
- 60. To Prevent Baking Ovens from Scorching A good method to prevent baking ovens from scorching or burning pastry is to sprinkle a mixture of sand and salt on the bottom where the pans are placed. This affords a way of radiating the heat evenly. The mixture also absorbs fruit juices, which may be spilled in the course of cooking. The covering is easily changed, which keeps the oven clean. The best proportion is half salt and half sand.
- 61. Horn Candle Sconce The person who cares for things unusual will find the candle sconce made of a cowhorn a suitable fixture for the den. A well shaped and not too large cowhorn is selected, and prepared by first partly filling it with paper, packed in tightly, then filling it to the top with plaster of Paris, in which a candle socket is formed. The bracket is made of strips of metal, formed as shown and riveted together where they touch each other, the back piece being fastened with screws to a wall board. The metal may be brass or copper and finished in nickel, antique, bronze, or given a brush finish. The wooden wall piece can be finished in any style desired. The Cowhorn with Bracket and Wall Board, Making an Unusual Candle Sconce for the Den
- 62. ¶ White spots on furniture can be removed by rubbing the wood with ammonia.
- 63. How to Make a Copper Stencil for Marking Laundry A stencil suitable for marking laundry may be easily made as follows: First procure a small sheet of stencil sheet copper, about 1 in. wide and 4 in. long. Dip this sheet of copper in a vessel containing some melted beeswax, so that both sides will be evenly covered with a thin coat of the wax when it cools. The design—name, monogram or figure—that is wanted in the stencil should now be drawn upon a piece of thin white paper, the reverse side of the paper blackened with graphite, and then laid on the stencil plate with the design in the center of the plate, whereupon the design is lightly traced with a blunt point on the thin wax coating. After the paper is removed, trace the design on the wax surface with a pointed instrument, but not completely, the lines being broken at more or less regular intervals, to form holders so that, after etching, the design cannot fall out. Next lay the stencil in a small shallow dish and pour a small quantity of fresh nitric acid over it. Keep the air bubbles removed from the surface by means of a piece of soft feather. The design will be eaten away in a very short time, where the wax has been removed, and this may be readily observed by holding the stencil plate up to the light. The acid should then be rinsed off with water, and the wax removed by heating and wiping it off with a cloth. The stencil may be given a final cleaning in a dish of benzine or gasoline, which will remove any remaining wax.
- 64. A Brass Pin Tray A novelty pin tray can be easily made of a piece of No. 24 gauge sheet brass or copper, 5 in. in diameter. The metal is annealed and polished with fine emery cloth, which is given a circular motion to produce a frosted effect. The necessary tools are a 1-in. hardwood board with a 2-1/2-in. hole bored in it, and a round piece of hard wood, 1-7/8 or 2 in. in diameter, with the ends sawn off square. The Former and Method of Using It to Produce a Wrinkled Edge on the Tray Place the sheet metal centrally over the hole in the board and set one end of the round stick in the center of the metal. Drive the stick with a hammer until a recess about 1 in. deep is made in the center. The edge of the metal will wrinkle up as shown in the sketch. It is
- 65. scarcely possible to make two trays alike, as the edge almost invariably will buckle in a different manner.—Contributed by F. Van Eps, Plainfield, N. J.
- 66. A Homemade Exerciser A weight machine for exercising the muscles of the arms is easily constructed by using two screw hooks, 5 in. long, and two small pulleys, 2-1/2 in. in diameter. An awning pulley can be used for this purpose. The hole at the top of the hanger will allow the pulley to freely turn at almost any angle. A paving brick or a piece of metal can be used as a weight for each rope.—Contributed by Sterling R. Speirs, St. Louis, Mo. The Yoke of the Pulley is so Arranged as to Make It Move in All Positions on the Hook
- 67. A Book Covering New books can be quickly and neatly covered to keep them clean by cutting a paper large enough to cover the back and sides when the book is closed, allowing 1 in. extra at each end to be turned over the front and back edges, then pasting on corners cut from used envelopes. The paper jacket can be slipped on or off easily when the book is opened, and it will keep a new cover clean while the book is being handled.—Contributed by Dr. John A. Cohalan, Philadelphia. Paper Covering Kept in Place with Corners Cut from Old Envelopes and Pasted on the Paper
- 68. A Tilting Inkstand An ink-bottle stand, that can be tilted or adjusted so that the pen will always be filled with a sufficient quantity of ink even when little of it remains in the bottle, as shown in the sketch, can be easily made by the amateur. The base may consist of a square piece of sheet brass, which has soldered or riveted to its center two pieces of spring brass, placed crosswise and bent upward so as to form clips to hold the bottle firmly. The legs are made of two lengths of wire, of sufficient stiffness, and are shaped to form holders for lead pencils and penholders. One pair of the legs may be soldered to the brass plate and the opposite side of the latter rolled over the other pair so as to allow them either to stand upright or be depressed in order to tilt the stand, when the ink supply in the bottle gets low. Tilting Stand for an Ordinary Ink Bottle to Give Access for a Small Supply of Ink
- 69. A Ring Trick The trick to be described is one of the simplest and at the same time one of the most effective, and but little make-ready is required to perform it. The magician, while sitting in a chair, allows his hands to be tied together behind the back of the chair. A ring is placed between his lips which he claims to be able to slip on his finger without untying his hands. This, to the audience, seems practically impossible, but it is easily accomplished. A screen is placed in front of the performer before the trick is started, so that the audience will not see how it is done. As soon as he is hidden from view, he tilts his head forward and drops the ring in his lap. He then allows the ring to drop to the seat of the chair between his legs. The chair is tilted backward slightly, and he raises himself to allow the ring to slip to the back part of the chair seat, where he catches it in his hands and slips it on the finger. Any one finger may be mentioned, as he can slip the ring as readily on one as on another. Use a leather-bottom chair, if possible, as the least noise will then be made when the ring is dropped.—Contributed by Abner B. Shaw, N. Dartmouth, Massachusetts.
- 70. Removing Old Putty A very effective way to remove old putty from window panes or other articles is to apply a red-hot iron, as follows: The iron should be made of a broken file or cold chisel and the point heated quite hot. This is run over the surface of the putty, which will crack and fall off. Be careful not to let the hot iron touch the glass, as the heat may cause the latter to break.
- 71. How to Make a Water Wheel The materials used in the construction of this water wheel are such as the average amateur mechanic may pick up or secure from a junk pile. The drawings in Fig. 1 clearly show the way the wheel is built. The nozzle, Fig. 2, is made of pipe and fittings and is adjustable to concentrate the stream so as to get the full efficiency of the weight and velocity of the water. The cap on the end of the nipple is drilled to receive the pin point filed on the end of the 1/4-in. rod. The parts of this nozzle are a 1/2-in. tee, connected to the source of water supply; a plug, drilled to snugly fit the 1/4-in. rod, and fitted into one end of the straight part of the tee; and a 1/2-in. nipple of sufficient length to make the dimension shown in the sketch. The nipple has a long thread to receive two 1/2-in. locknuts, which clamp the nozzle to the sheet-metal covering, as shown in Fig. 1. Details of the Water Wheel (Fig. 1) The buckets, Fig. 3, are formed of some easily melted, but not too soft metal alloy which can be cast in plaster molds. They are attached with rivets to the circumference of 1/16-in. thick sheet-
- 72. metal disk of the diameter given in Fig. 1. This disk is fastened to a 1/4-in. shaft, 6 in. long, with two collars, one on each side of the disk, both being riveted to the disk and pinned to the shaft. The bearings AA are made of 3/4-in. pipe, each 2-1/4 in. long. Long threads are cut on these to turn through the two 3/4-in. waste nuts BB, which provides a way to adjust the buckets centrally with the stream of water, and to take up any side motion. The pipe is babbitted and drilled for oil holes. The runner or wheel must be well balanced, as the speed will be from 2,000 to 2,500 revolutions per minute with ordinary city pressure. In balancing the wheel, instead of adding an extra weight, a part of the disk is filed out on one edge. The inclosing sides are made of wood—cypress preferred—having the dimensions given, and two 7/8 by 1-1/2-in. pieces are attached to the bottom outside surfaces for mounting the wheel. The curved part is covered with galvanized sheet metal. (Fig. 2, Fig. 3)
- 73. The drawing shows a wheel of small diameter, but having considerable power. Greater power may be obtained by increasing the size of the jet and the diameter of the wheel, but the use of too many buckets results in decrease of power. One bucket should be just entering the stream of water, when the working bucket is at a point at right angles to the stream. The water should divide equally exactly on the center of the bucket and get out of the way as soon as possible. Any stagnant water in the case, or dead water in the bucket, is detrimental to the power. A free exit for the water is made at the bottom of the case, as shown. Metal Casing Instead of Wood (Fig. 4) The construction of the case may be varied and, instead of wood, metal sides and frame may be used. Where the builder cares to make a more substantial wheel and has access to a foundry, the metal parts can be made as shown in Fig. 4. The parts are in this instance fastened together with machine screws. Patterns are made and taken to a foundry for the castings, which are then machined to have close fitting joints.—Contributed by R. H. Franklin, Unnatosa, Wis.
- 74. An Interesting Experiment Take an ordinary board, 2 or 3 ft. long, such as a bread board, and place it on the table so that about one-third of its length will project over the edge. Unfold a newspaper and lay it on the table over the board as shown in the sketch. Anyone not familiar with the experiment would suppose the board could be knocked off by hitting it on the outer end. It would appear to be easy to do, but try it. Unless you are prepared to break the board you will probably not be able to knock the board off. Striking the Board The reason is that when the board is struck it forces the other end up and the newspaper along with it. This causes a momentary vacuum to be formed under the paper, and the pressure of the air above, which is about 15 lb. to the square inch, prevents the board from coming up. This is an entertaining trick to play at an evening party, and also makes a simple and interesting school experiment.
- 75. Ironing-Board Holder An ironing board that had been used on two chairs was cut off square on one end and a piece of heavy sheet metal cut and bent into the shape shown in Fig. 1. The square end of the board was fitted into the socket formed by the sheet metal. After attaching the socket to the wall with screws the board was easily put in place as shown in Fig. 2. The brace is hinged to the under side of the board. —Contributed by L. G. Swett, Rochester, N. Y. Socket and Manner of Holding Board (Fig. 1, Fig. 2)
- 76. How to Make a Water Motor By Edward Silja After making several different styles of water motors I found the one illustrated to be the most powerful as well as the simplest and most inexpensive to make. It can be constructed in the following manner: A disk, as shown in Fig. 1, cut from sheet iron or brass, 1/16 in. thick and 9-3/4 in. in diameter, constitutes the main part of the wheel. The circumference is divided into 24 equal parts, and a depth line marked which is 8-1/4 in. in diameter. Notches are cut to the depth line, similar to the teeth of a rip saw, one edge being on a line with the center of the wheel and the other running from the top of one tooth to the base of the preceding tooth. Metal Disk with a Saw-Tooth Circumference That Constitutes the Main Body of the Wheel (Fig. 1) A 1/4-in. hole is drilled in the center of the disk and the metal strengthened with a flange, placed on each side of the disk and
- 77. fastened with screws or rivets. A 1/4-in. steel rod is used for the shaft. The cups, or buckets, are shaped in a die which can be cast or built up of two pieces, as desired. Both of these dies are shown in Fig. 2. The one at A is made of two pieces riveted together. Two Ways of Making the Dies to Shape the Sheet-Metal Water Cups (Fig. 2) If a foundry is near, a pattern can be made for a casting, as shown at B. The die is used in the manner shown in Fig. 3. A strip of galvanized metal is placed over the depressions in the die and a ball- peen hammer used to drive the metal into the die. Cups, or buckets, are thus formed which are soldered to the edge of the teeth on a line with the center of the disk, as shown in Fig. 4. As there are 24 notches in the disk, 24 cups will be necessary to fill them.
- 78. The Sheet Metal is Placed on the Die and Then Hammered into Shape (Fig. 3) The cups are made in pairs or in two sections, which is a better construction than the single cup. The water from the nozzle first strikes the center between the cups, then divides and produces a double force. The Water Cups are Fastened to the Teeth on the Metal Disk with Solder (Fig. 4) When this part of the work is finished it is well to balance the wheel, which can be done by filing off some of the metal on the heavy side or adding a little solder to the light side. This will be necessary to provide an easy-running wheel that will not cause any unnecessary wear on the bearings. The housing for the wheel consists of two wood pieces, about 3/4-in. thick and cut to the shape shown in Fig. 5. Grooves are cut in one surface of each piece, to receive the edges of a strip of galvanized metal, as shown at A. The grooves are cut with a specially constructed saw, shown in Fig. 6. It consists of a piece of wood, 6 in. long, 1-1/2 in. wide and 1/2 in. thick, the end being cut
- 79. on an arc of a circle whose diameter is 10 in. A piece of a broken hacksaw blade is fastened with screws to the curved end. A nail is used as a center pivot, forming a 5-in. and a 5-3/4-in. radius to swing the saw on in cutting the groove. After inserting the strip of galvanized metal, A, Fig. 5, the sides are clamped together with bolts about 3-1/4 in. long. The Housing for the Wheel with a Connection to Attach the Motor on an Ordinary Faucet (Fig. 5) Construction of the Saw for Making the Groove to Receive the Metal Strip in the Sides (Fig. 6) A piece of pipe, B, Fig. 5, having an opening 3/8-in. in diameter, is soldered onto the metal strip A. An ordinary garden-hose coupling, C, is soldered to the end of the pipe. A bearing, D, shaped as shown, is fastened to one of the wood sides with screws, the wheel shaft is run into it, and the parts assembled. A wheel, either grooved or flat, 2-1/2 or 3 in. in
- 80. diameter, is placed on the shaft. The hose coupling makes it easy to connect the motor directly to the water faucet.
- 81. An Application for Small Wounds Pure wintergreen oil makes a good local application for all small wounds, bites, scratches, abrasions, etc. There is no germ or microbe, animal or vegetable, dead or living, that can withstand this oil, and at the same time it is not injurious to living tissues. A few drops gently rubbed in where there is apt to be any infection is sufficient. An infection always follows the wound of a bullet or the scratch of a brass pin, with irritation extending up the limb or part threatening tetanus or lockjaw. These symptoms are manifested by spasmodic pains which shoot upward, but are quickly subdued, if the oil is applied along the track of the pain or infection. This oil is equally effective when locally applied to tendons or ligaments which have been unduly strained. An ounce of the pure oil does not cost much, and it should be kept in every shop and household. If 5 or 10 per cent of olive oil is added to it, the oil will have more body and will last longer.— Contributed by Dr. E. R. Ellis, Detroit, Mich.
- 82. Cores for Use in Babbitt Metal It is often necessary in making things of babbitt metal to core out some of the parts. A very good core is made of common salt and glue. Mix just enough of the glue into the salt to make a stiff paste, which is then formed into the desired shape or molded in a core box and allowed to harden. This kind of a core can be removed from the casting by soaking it in warm water, which will dissolve the salt and leave the desired hole.—Contributed by H. F. Hopkins, N. Girard, Pa.
- 83. How to Build a Wind Vane with an Electric Indicator Quite often it is practically impossible to ascertain the direction of the wind by observing an ordinary wind vane on account of the necessity of locating the vane at such a height that it may give a true indication. By means of the device shown in Fig. 2, the position of the vane may be determined without actually looking at the vane itself and the indicating device may be located almost anywhere and independently of the position of the wind vane. Fig. 1—The Diagram of a Wheatstone Bridge Which Shows the Points of Contact So Placed That a Balance is Obtained The principle upon which the device operates is that of the Wheatstone bridge. The position of the moving contact A, Fig. 1, is controlled by the wind vane. This contact is made to move over a specially constructed resistance R, Fig. 2. A second movable contact, B, is controlled by the observer and moves over a second resistance, identical with that over which the contact A moves. These two resistances are connected so as to form the two main branches of a Wheatstone bridge; the points A and B are connected to the current-
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