![Exploiting Chaos for Quantum Control
Ying-Cheng Lai
Abstract The field of Quantum Chaos is referred to as the study of quantum
behaviors of systems whose corresponding classical dynamics are chaotic, or study
of quantum manifestations of classical chaos. Equivalently, it means that quantum
behaviors depend on the nature of the classical dynamics, implying that classical
chaos can be used to control or manipulate quantum behaviors. We discuss two
examples here: using transient chaos to control quantum transport in nanoscale sys-
tems and exploiting chaos to regularize relativistic quantum tunneling dynamics in
Dirac fermion and graphene systems.
1 Introduction
Controlling chaos in dynamical systems has been studied for more than two decades
since the seminal work of Ott, Grebogi, and Yorke [1]. The basic idea was that chaos,
while signifying random or irregular behavior, should not be viewed as a nuisance
in applications of nonlinear dynamical systems. In particular given a chaotic system,
the fact that there are an infinite number of unstable periodic orbits embedded in the
underlying chaotic invariant set means that there are an equally infinite number of
choices for the operational state of the system depending on need, provided that any
such state can be stabilized. Then, the intrinsically sensitive dependence on initial
conditions, the hallmark of any chaotic system, implies that it is possible to apply
small perturbations to stabilize the system about any desirable state. Controlling
chaos has since been studied extensively and examples of successful experimental
implementation abound in physical, chemical, biological, and engineering systems
Y.-C. Lai (B)
School of Electrical, Computer and Energy Engineering, Arizona State University,
Tempe, Arizona 85287, USA
e-mail: Ying-Cheng.Lai@asu.edu
V. In et al. (eds.), International Conference on Theory and Application in Nonlinear 3
Dynamics (ICAND 2012), Understanding Complex Systems,
DOI: 10.1007/978-3-319-02925-2_1, © Springer International Publishing Switzerland 2014](https://image.slidesharecdn.com/639-250309191420-78e9f939/85/International-Conference-on-Theory-and-Application-in-Nonlinear-Dynamics-ICAND-2012-1st-Edition-Ying-Cheng-Lai-Auth-17-320.jpg)
![4 Y.-C. Lai
[2]. The vast literature on controlling chaos, however, has been limited to nonlinear
dynamical systems in the classical domain.
Recently, it has been articulated that chaos can be exploited to control or manip-
ulate quantum-mechanical behaviors [3, 4]. For example, in the context of quantum
transport through nanostructures, a fundamental characteristic is conductance fluc-
tuations. It has been demonstrated that chaos, more specifically transient chaos,
can be effective in modulating the conductance-fluctuation patterns, and it has been
suggested [4] that this can be realized experimentally by applying an external gate
voltage in a device of suitable geometry to generate classically inaccessible poten-
tial barriers. Adjusting the gate voltage allows the characteristics of the dynamical
invariant set responsible for transient chaos to be varied in a desirable manner which,
in turn, can induce continuous changes in the statistical characteristics of the quantum
conductance-fluctuation pattern. In another example, it has been shown that chaos
can be utilized to suppress, significantly, the spread in the tunneling rate commonly
seen in systems whose classical dynamics are regular, and this is called regularization
of quantum tunneling dynamics by chaos [3]. More recently, it has been demonstrated
that similar effects arise in relativistic quantum systems [5].
This Brief Review has two purposes: (1) to discuss the two aforementioned
examples of exploiting chaos for quantum control, and (2) to argue that the prin-
ciple of chaos-based quantum control is more general with potentially significant
applications in nanoscience.
2 Controlling Quantum Transport by Transient Chaos
A fundamental quantity characterizing the transport of an electron through a nanos-
tructure, such as a quantum dot or a quantum point contact, is quantum transmission
probability, or simply quantum transmission. In general, quantum transmission is
determined by many electronic and system parameters such as the Fermi energy, the
strength of external magnetic field (if there is one), and the details of the geometry of
the structure. If the structure is connected through electron waveguides (or leads) to
electron reservoirs (i.e., contacts) to form a circuitry, the conductances defined with
respect to various voltage biases among the contacts, together with the correspond-
ing currents, will be determined by the quantum transmission [6]. This means that
the conductances can also depend sensitively on electronic and geometrical para-
meters. For example, as the Fermi energy of the electron changes, the conductances
can exhibit wild fluctuations and sharp resonances [7–13]. In applications such as
the development of electronic circuits and nanoscale sensors, severe conductance
fluctuations are undesirable and are to be eliminated so that stable device operation
can be achieved. The outstanding question is then, can practical and experimentally
feasible schemes be articulated to modulate quantum conductance fluctuations? It
has been demonstrated recently and understood theoretically that classical transient
chaos can be used to effectively modulate conductance-fluctuation patterns associ-
ated with quantum transport through nanostructures [4].](https://image.slidesharecdn.com/639-250309191420-78e9f939/85/International-Conference-on-Theory-and-Application-in-Nonlinear-Dynamics-ICAND-2012-1st-Edition-Ying-Cheng-Lai-Auth-18-320.jpg)
![Exploiting Chaos for Quantum Control 5
Intuitively, the basic principle underlying the transient-chaos based strategy for
modulating quantum transport is quantum interference. It has been known that quan-
tum pointer states, which are resonant states of finite but long lifetime formed inside
the nanostructure [14–16], can cause sharp conductance fluctuations - a kind of Fano
resonance [17, 18]. To give a specific example, consider quantum-dot systems, a par-
adigm for investigating all kinds of quantum transport phenomena through nanostruc-
tures. Such a system typically consists of a finite device region of certain geometrical
shape, such as a square, a circle, or a stadium, and a number of leads connected with
the device region. For a quantum-dot system whose classical dynamics is either regu-
lar or contains a significant regular component, there are stable periodic orbits in the
classical limit. If the dot geometry is closed, highly localized states can form around
the classically stable periodic orbits as a result of quantum interference. When elec-
tronic waveguides (leads) are attached to the quantum dot so that it is open, some
periodic orbits can still survive, leading to resonant states, or quantum pointer states.
Since the corresponding classical orbits are stable, the resonant states can have long
lifetime, so their coupling to the leads is weak. As a result, narrow resonances can
form around the energy values that are effectively the eigenenergies for the stable pe-
riodic orbits in the corresponding closed system. When the dot geometry is modified
so that the underlying classical dynamics becomes fully chaotic, no stable periodic
orbits can exist. Although scars can still form around classically unstable periodic
orbits in a closed chaotic system [19], the corresponding resonant states in the open
system generally will have much shorter lifetimes. This means that these resonant
states do couple to the leads more strongly, broadening the narrow resonances in
the conductance-fluctuation pattern. Here chaos is transient because the system is
open. According to the theory of transient chaos [20], the dynamical invariant sets
responsible for transient chaos are non-attracting chaotic sets in the phase space. If
the properties of transient chaos can be adjusted experimentally by parameter tuning,
the quantum conductance fluctuation-patterns can then be controlled in a desirable
manner. For example, one can change the effective geometry of the dot structure con-
tinuously so as to enhance the escape rate, a basic quantity characterizing transient
chaos, and this could lead to significantly smoother quantum-conductance fluctua-
tions.
To realize quantum control by using chaos, we conceive generating a region about
the center of the device or structure with high potential so that it is impenetrable to
classical particles. For example, consider a square quantum dot, a prototypical model
in semiconductor two-dimensional electron-gas (2DEG) systems. When the dot is
closed, the corresponding classical dynamics is integrable so that extremely nar-
row resonances can arise in the quantum transport dynamics of the corresponding
open-dot system. Now imagine applying a gate voltage perpendicular to the device
plane to generate a circular, classically forbidden region about the center of the dot,
as shown schematically in Fig.1. In general, the potential profile will be smooth in
space. However, qualitatively, the classical scattering behavior is similar to that from
an infinite potential well. Thus it is reasonable to impose the infinite potential-well
assumption for the central region, which defines a “forbidden” region. Varying the
voltage V0 can change the effective radius R of the forbidden region. Classically, the](https://image.slidesharecdn.com/639-250309191420-78e9f939/85/International-Conference-on-Theory-and-Application-in-Nonlinear-Dynamics-ICAND-2012-1st-Edition-Ying-Cheng-Lai-Auth-19-320.jpg)
![6 Y.-C. Lai
R
Left Lead Right Lead
Fig. 1 Illustration of a possible scheme to control transport through a two-dimensional quantum-dot
system. When semiconductor materials (e.g., silicon) are used, the system is the traditional two-
dimensional electron-gas (2DEG) system described by the Schrödinger equation in non-relativistic
quantum mechanics. If the material is graphene, in certain energy regime the system is described
by the Dirac equation in relativistic quantum mechanics. By applying a suitable gate voltage per-
pendicular to the device plane to generate a circular forbidden region at the center of the device,
the resulting closed system is a Sinai billiard, whose classical dynamics is fully chaotic. Open
quantum-dot system can be formed by attaching two leads to the billiard system, one on the left and
another on the right side. The classical dynamics of the device can thus be characterized as chaotic
scattering
closed system is thus a Sinai billiard [21], which is fully chaotic, insofar as the radius
of the central potential region R is not zero. When leads are connected to the device
region so as to open the system, chaos becomes transient. The dynamical character-
istics of the underlying chaotic invariant set can be adjusted in a continuous manner
by increasing the radius R [22]. Quantum mechanically we thus expect to observe
increasingly smooth variations in the conductance with, e.g., the Fermi energy, which
has been demonstrated [4] using both semiconductor 2DEG and graphene [23–26]
systems.
Insights into why classical chaos can smooth out quantum conductance fluctua-
tions can also be gained from the semiclassical theory of quantum chaotic scattering
[27–29]. In particular, in the semiclassical regime, it was established by Blümel
and Smilansky that the energy autocorrelation function of the quantum transmission
fluctuation is proportional to the Fourier transform of the particle-decay law in the
classical limit [27]. For fully developed chaotic transport through a quantum dot, the
decay law is exponential with the rate κ. As a result, the quantum energy correlation
function decays as a Lorentzian function with the width given by κ, where is the
Planck’s contant. In the theory of transient chaos [20], κ is the escape rate associated
with the underlying non-attracting chaotic set. As the radius of the central potential
region is increased, κ also increases. The energy autocorrelation function then decays
more slowly, signifying less fluctuations, i.e., less number of sharp resonances in
the quantum transmission. This semiclassical argument suggests that the degree of
quantum transmission fluctuations can be controlled by classical chaos.
Extensive numerical support for the principle of transient-chaos based control of
quantum transport and a detailed theoretical analysis can be found in Ref. [4, 30].](https://image.slidesharecdn.com/639-250309191420-78e9f939/85/International-Conference-on-Theory-and-Application-in-Nonlinear-Dynamics-ICAND-2012-1st-Edition-Ying-Cheng-Lai-Auth-20-320.jpg)
![Exploiting Chaos for Quantum Control 7
3 Effect of Chaos on Quantum Tunneling
The principle of regularization of quantum tunneling by chaos can be understood
by considering the prototypical system in Fig.2, which consists of two symmetrical
cavities connected by a one-dimensional potential barrier along the line of symmetry.
When the classical dynamics in each cavity is integrable, for sufficiently large energy
the tunneling rate can have many values in a wide interval. Choosing the geometry
of the cavity such that the classical dynamics become chaotic can greatly enhance
and regularize quantum tunneling. Heuristically, this can be understood, as follows.
When the potential barrier is infinite, each cavity is a closed system with an infinite
set of eigenenergies and eigenstates. Many eigenstates are concentrated on classical
periodic orbits, forming quantum scars [19]. For a classically integrable cavity, some
stable or marginally stable periodic orbits can persist when the potential barrier
becomes finite so that each cavity system is effectively an open quantum system.
Many surviving eigenstates correspond to classical periodic orbits whose trajecto-
ries do not encounter the potential barrier, generating extremely low tunneling rate
even when the energy is comparable with or larger than the height of the potential
barrier. The eigenstates corresponding to classical orbits that interact with the poten-
tial barrier, however, can lead to relatively strong tunneling. In a small energy interval
the quantum tunneling rate can thus spread over a wide range. However, when the
classical dynamics is chaotic, isolated orbits that do not interact with the potential
barrier are far less likely and, consequently, the states associated with low tunneling
rates disappear, effectively suppressing the spread in the tunneling rate.
The idea of using chaos to regularize quantum tunneling dynamics was first
conceived and demonstrated in non-relativistic quantum systems governed by the
Schrödinger equation [3]. Recently, the question of whether chaos can regularize
tunneling in relativistic quantum systems has been addressed [5], where the motion
of massless Dirac fermions in the setting of resonant tunneling was investigated to
facilitate comparison with the non-relativistic quantum case. In general, it is a chal-
Fig. 2 A closed quantum
system of arbitrary shape in
two dimensions
Potential barrier](https://image.slidesharecdn.com/639-250309191420-78e9f939/85/International-Conference-on-Theory-and-Application-in-Nonlinear-Dynamics-ICAND-2012-1st-Edition-Ying-Cheng-Lai-Auth-21-320.jpg)
![8 Y.-C. Lai
lenging task to solve the Dirac equation in a confined geometry, due to the difficulties
to incorporate zero-flux boundary conditions and to remove artificial, non-physical
effects such as fermion doubling as a result of spatial discretization. A numerical
scheme has then been developed [31] to overcome these difficulties, which is based
on constructing a physically meaningful, Hermitian Hamiltonian. Extensive compu-
tations have revealed unequivocally the existence of surviving eigenstates that lead
to extremely low tunneling rates. As for the non-relativistic quantum case, making
the cavities classically chaotic can greatly regularize the quantum tunneling dynam-
ics. To explore the practical implications, resonant tunneling devices made entirely
of graphene [23] have been studied [5], where the tunneling rates for different en-
ergy values have been calculated. Qualitatively similar results have been obtained to
those for massless Dirac fermions. One unique feature for both the Dirac fermion and
graphene systems, which finds no counterpart in non-relativistic quantum tunneling
devices, is the high tunneling rate in the regime where the particle energy is smaller
than the height of the potential barrier. This is a manifestation of the Klein-tunneling
phenomenon [32–34]. A theory has been developed [5] to explain the numerical
findings, which is based on the concept of self energies and the complex energy
spectrum of the non-Hermitian Hamiltonian for the “open” cavity.
4 General Thoughts on Chaos-Based Quantum Control
The general principle of chaos-based control of quantum behaviors is motivated by
the term quantum chaos, which does not mean that there can actually be chaos in
quantum mechanical systems but is referred to as the study of quantum manifestations
of systems whose dynamics in the classical limit exhibit chaos [35, 36]. The basic
reason that chaos may be ruled out in quantum systems is that the fundamental
governing equations, the Schrödinger equation or the Dirac equation, are linear. At
the present, there is tremendous literature on quantum chaos, where various quantum
manifestations of classically chaotic systems have been studied. The general result is
that distinct classical dynamics, integrable or chaotic, can lead to characteristically
different quantum behaviors. Furthermore, different types of chaotic behaviors can
generate distinct quantum manifestations. From the point of view of control, all
these suggest that quantum behaviors can be manipulated or harnessed for desirable
applications by choosing distinct classical dynamical behaviors, in particular chaotic
dynamics.
The two examples discussed in this Brief Review, control of quantum transport
and quantum tunneling, are based on building chaos into the system. In the transport
problem that involves the Sinai billiard type of device structure, the properties of
the underlying chaotic set can be modified, for example, by an externally adjustable
gate voltage. In the tunneling problem, the geometry of the cavities are deliberately
designed to yield chaotic dynamics in the classical limit. Once the structure is fixed,
experimentally it may be difficult to change the characteristics of chaos. It is thus nec-](https://image.slidesharecdn.com/639-250309191420-78e9f939/85/International-Conference-on-Theory-and-Application-in-Nonlinear-Dynamics-ICAND-2012-1st-Edition-Ying-Cheng-Lai-Auth-22-320.jpg)
![Exploiting Chaos for Quantum Control 9
Fig. 3 A four terminal device
in the presence of magnetic
field. The area enclosed by the
greenlines indicatesthe region
in the physical space where
transient chaos can occur. The
black circle marks the region
in which the magnetic field is
applied
essary to search for experimentally feasible schemes to modulate the characteristics
of the underlying chaotic invariant sets in a continuous fashion.
Figure3 presents a possible scheme where a single external parameter can be var-
ied to realize chaos-based control of quantum transport. It is a four-terminal device,
where four idealized leads join smoothly to form a quantum-dot structure, which has
been used widely in the study of, for example, quantum Hall effect [6]. The struc-
ture typically exhibits chaotic scattering (transient chaos) in the classical limit. A
perpendicular magnetic field can be applied. An earlier work [37] demonstrated that
the dynamical invariants of the underlying non-attracting chaotic set can be modi-
fied continuously by changing the strength of the magnetic field. It is thus possible
to modulate the quantum conductance-fluctuation patterns by simply adjusting the
magnetic-field strength [38].
Acknowledgments The main idea of using chaos to manipulate quantum behaviors was generated
through extensive discussions with Dr. L. Pecora from Naval Research Laboratory in January 2011
at Dr. M. Shlesinger’s ONR Program Review Meeting at UCSD. The computations and theoretical
analyses reported in the references [4, 5, 30, 31] on which this Review is based were mainly carried
out by Dr. R. Yang, Dr. X. Ni, and Dr. L. Huang, all formerly affiliated with ASU.
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18. H. Ishio, Phys. Rev. E 62, R3035 (2000)
19. Given a closed Hamiltonian system that exhibits fully developed chaos in the classical limit,
one might expect the quantum wavefunctions associated with various eigenstates to be more or
less uniform in the physical space. However, in the seminal work of McDonald and Kaufman
[Phys. Rev. Lett. 42, 1189 (1979) and Phys. Rev. A 37, 3067 (1988)], it was observed that
quantum eigen-wavefunctions can be highly non-uniform in the chaotic stadium billiard. A
systematic study was subsequently carried out by Heller [Phys. Rev. Lett. 53, 1515 (1984)],
who established the striking tendency for wavefunctions to concentrate about classical unstable
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wavefunctions are generally not expected to be uniform. Relativistic quantum scars in chaotic
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and R. Akis, Phys. Rev. Lett. 103, 054101 (2009)].
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31. X. Ni, L. Huang, Y.-C. Lai, C. Grebogi, Phys. Rev. E 86, 015702 (2012)
32. P. Strange, Relativistic Quantum Mechanics with Applications in Condensed Matter Physics
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35. F. Haake, Quantum Signatures of Chaos, 2nd edn. (Springer, Berlin, 2001)
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38. G.-L. Wang, L. Ying, Y.-C. Lai, and C. Grebogi, Phys. Rev. E 87, 052908 (2013)](https://image.slidesharecdn.com/639-250309191420-78e9f939/85/International-Conference-on-Theory-and-Application-in-Nonlinear-Dynamics-ICAND-2012-1st-Edition-Ying-Cheng-Lai-Auth-24-320.jpg)
![16 K. Onu et al.
If ξ is white noise we can solve for η explicitly. Its power spectral density is
Sη(ω) =
ε2ν2S0
(1 − ω2)2 + 4ε4ζ2
o ω2
,
where S0 is the power spectral density of ξ. The peak intensity and the carrying
frequency of η are determined by the filter parameter ζo.
The stability of the locked mass steady-state oscillation is now obtained by using
the first-order approximation of sine and cosine in the dynamics for θ. We get
θ̈ + 2ε2
ζpθ̇ + ((q0 + ε2
μ)2
+ εη̈)θ = 0,
and the power spectral density of η̈ is given by
Sη̈(ω) =
ω4ε2ν2S0
(1 − ω2)2 + 4ε4ζ2
o ω2
.
The maximal Lyapunov exponent can now be easily calculated and the stability
boundary can be obtained in terms of excitation intensity ν and the dissipation coef-
ficients ζp. An explicit expression for the maximal Lyapunov exponents of the single
mode solution is given by expanding it in ε, we have
λ1 ≈ ε2
−ζp +
1
8 q2
o
Sη̈(2 (qo + ε2
μ))
and λ2 = ε2
−ζp −
1
8 q2
o
Sη̈(2 (qo + ε2
μ))
.
The noise has no effect on the other two exponents; i.e., λ3 = λ4 = −ε2ζo.
Since the point θ ≡ 0 is a stable point for the hanging pendulum, the pendulum
undergoes small random motion near θ ≡ 0, and all four Lyapunov exponents
are negative. However, as we further increase the noise intensity, the top exponent
becomes positive when ν2S0 = 8ζ2
o ζp. The system then becomes unstable, and the
following question arises.
• Do both the mass spring oscillator and the pendulum undergo random vibrations
when the top exponent becomes positive (i.e., ν2 S0 8 ζ2
o ζp), i.e., does a new
coupled-mode “stationary solution” or “stationary density function” appear?
3 Coupled Mode Solutions
Making use of a time-varying symplectic transformation (see [1] for details), we
arrive at
ẋε
t = εb1
(xε
t , t) + ε2
b2
(xε
t , t : ζ, μ) + εσ(xε
t , t : ν)ξ(t), (4)
where(x1, x2)and(x3, x4)areconjugatepairsandcanbethoughtofastheamplitudes
of periodic orbits of the dominant dynamics.](https://image.slidesharecdn.com/639-250309191420-78e9f939/85/International-Conference-on-Theory-and-Application-in-Nonlinear-Dynamics-ICAND-2012-1st-Edition-Ying-Cheng-Lai-Auth-30-320.jpg)
![Random Vibration of a Nonlinear Autoparametric System 17
The coefficients b1, b2, σ are periodic in time. Standard deterministic averaging
can be used to average out the effects of rapidly-oscillating periodic coefficients. Let
M be this averaging operator.
Definition 1 (Time averaging operator) For a function ϕ ∈ C∞(R4 × R) which is
2π periodic in its last argument, define the time averaging operator M by
(Mϕ)(x) ≡
1
2π
2π
0
ϕ(x, t)dt.
From the explicit formulas (see [1]) for b1 (where q = 1/2), we see that for x =
(x1, x2, x3, x4) ∈ R4,
(Mb1
)(x) =
−1
2 x2x4, 1
2 (x1x4 − x2x3), 1
4 (x2
2 − x2
4 ), 1
2 (x1x2 + x3x4) .
Then the averaged system ẋt = (Mb1)(x) is a 4-dimensional Hamiltonian system
with two first integrals K and I in involution.
The Hamiltonian associated with these dynamics is
K(x) =
1
4
x1(x2
4 − x2
2 ) −
1
2
x2x3x4. (5)
The unperturbed four-dimensional Hamiltonian system
ż = ¯
∇K(z) (6)
has two first integrals in involution, namely, the Hamiltonian itself (5) and a second
constant of motion (momentum variable)
I (x) = (x2
1 + x2
3 ) +
1
2
(x2
2 + x2
4 ). (7)
The invariant I is functionally independent of K, exists globally and is single
valued. Note that the Hamiltonian system’s equations remain unchanged when
t → −t, x1 → −x1 and x3 → −x3.
3.1 Dimensional Reduction
Our main analytical tool is a certain method of dimensional reduction of nonlinear
systems with symmetries and small noise. As the noise becomes asymptotically
small, one can exploit symmetries and a separation of scales to use well-known
methods (viz. stochastic averaging) to find an appropriate lower-dimensional
description of the system.](https://image.slidesharecdn.com/639-250309191420-78e9f939/85/International-Conference-on-Theory-and-Application-in-Nonlinear-Dynamics-ICAND-2012-1st-Edition-Ying-Cheng-Lai-Auth-31-320.jpg)
![Random Vibration of a Nonlinear Autoparametric System 19
The relation between K, u1, u2, and I is illustrated in Fig.2. Note that this system’s
equations remain unchanged when t → −t, u2 → −u2 and ψ → −ψ. System (11)
has four fixed points. They are (u1, u2) = (0, ±
√
I) and (u1, u2) = (±
√
3I
3 , 0).
The points on the u1 axis are saddle points and those on the u2 axis are center
fixed points.
3.1.2 State Space of Yε
t
The slow variable Yε
t evolves on an arrowhead. Let S
def
={x ∈ R4 : K∗ K(x)
K∗, 0 I (x) I∗}. Then define an equivalence relation ∼ on R4 by identifying
x ∼ y if x and y are on the same orbit of the hamiltonian flow ẋt = (Mb1)(x).
Define M
def
= S̄/ ∼, and endow M with the quotient topology defined by ∼. If x ∈ S̄,
we let [x] := {y ∈ S̄ : y ∼ x} be the equivalence class of x. π(x) := [x]. The slow
variable Yε
t evolves on M = 2
i=1 Γi ∪ 2
i=0[ci ] ∪ 2
i=1 i where ci are the fixed
points, the i are closed orbits whose union is ∂S̄, and each Γi is the π-image of a
maximal open subset of R4 which does not intersect any of the [ci ] or i . The state
space is illustrated in Fig.3.
3.1.3 M A Averaging
If the external noise ξ(τ) represents mean zero, stationary, independent stochas-
tic processes with the strong mixing property, then roughly, as ε → 0, 1
ε ξ(t/ε2)
approaches a white noise process. Khasminskii [2] gave a rigorous proof that a fam-
ily of processes Xε
t converges to a diffusion process. The aim here is to make use of
this and derive a reduced graph-valued process for the integrals of motion, Yε.
We have pointed out that that there are three time-scales involved in our averaging
problems. The first step is to average the periodic fluctuations of the coefficients and
obtain M-averaged quantities as the precursors to the stochastically averaged drift
and diffusion coefficients. Somewhat laborious calculations yield the M-averaged
quantities
mi (x) ≡
M
F2
1 + f1 + g1
(x) and ai j (x) ≡
M
σσT
i j
(x). (13)
These calculations can be simplified by considering the symplectic transformation
(10) which provides a convenient geometric structure of the unperturbed integrable
Hamiltonian problem. In (K, I, u) coordinates, the drift and diffusion (13) coeffi-
cients are
m1(K, I, u) = −(ζo + 2ζp)K −
1
4
(8μ + 3I) K
u2
u1
+
1
2
3 +
1
R
K2 u2
u2
1
,
m2(K, I, u) = 2[σ2
Sξξ (1) − ζo I + 2(ζo − ζp)K/u1],
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- 5. Understanding Complex Systems Visarath In Antonio Palacios Patrick Longhini Editors International Conference on Theory and Application in Nonlinear Dynamics (ICAND 2012)
- 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 successfully 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 automata, adaptive systems, genetic algorithms and compu- tational intelligence. The three major book publication platforms of the Springer Complexity program are the monograph series ‘‘Understanding Complex Systems’’ focusing on the various applications of complexity, the ‘‘Springer Series in Synergetics’’, which is devoted to the quantitative the- oretical and methodological foundations, and the ‘‘SpringerBriefs in Complexity’’ which are concise and topical working reports, case-studies, surveys, essays and lecture notes of rele- vance to the field. In addition to the books in these two core series, the program also incor- porates individual titles ranging from textbooks to major reference works. Editorial and Programme Advisory Board Henry Abarbanel, Institute for Nonlinear Science, University of California, San Diego, USA Dan Braha, New England Complex Systems Institute and University of Massachusetts, Dartmouth, USA Péter Érdi, Center for Complex Systems Studies, Kalamazoo College, USA and Hungarian Academy of Sciences, Budapest, Hungary Karl Friston, Institute of Cognitive Neuroscience, University College London, London, UK Hermann Haken, Center of Synergetics, University of Stuttgart, Stuttgart, Germany Viktor Jirsa, Centre National de la Recherche Scientifique (CNRS), Université de la Méditerranée, Marseille, France Janusz Kacprzyk, System Research, Polish Academy of Sciences, Warsaw, Poland Kunihiko Kaneko, Research Center for Complex Systems Biology, The University of Tokyo, Tokyo, Japan Scott Kelso, Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, USA Markus Kirkilionis, Mathematics Institute and Centre for Complex Systems, University of Warwick, Coventry, UK Jürgen Kurths, Nonlinear Dynamics Group, University of Potsdam, Potsdam, Germany Andrzej Nowak, Department of Psychology, Warsaw University, Poland Linda 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, System Design, ETH Zurich, Zürich, Switzerland Didier Sornette, Entrepreneurial Risk, ETH Zurich, Zürich, Switzerland Stefan Thurner, Section for Science of Complex Systems, Medical University of Vienna, Vienna, Austria
- 7. Understanding Complex Systems For further volumes: http://www.springer.com/series/5394 Founding Editor: Scott Kelso Future scientific and technological developments in many fields will necessarily depend upon coming to grips with complex systems. Such systems are complex in both their composition—typically many different kinds of components interacting simultaneously and nonlinearly with each other and their environments on multiple levels—and in the rich diversity of behavior of which they are capable. The Springer Series in Understanding Complex Systems series (UCS) promotes new strategies and paradigms for understanding and realizing applications of complex systems research in a wide variety of fields and endeavors. UCS is explicitly transdisciplinary. It has three main goals: First, to elaborate the concepts, methods and tools of complex systems at all levels of description and in all scientific fields, especially newly emerging areas within the life, social, behavioral, economic, neuro- and cognitive sciences (and derivatives thereof); second, to encourage novel applications of these ideas in various fields of engineering and computation such as robotics, nano-technology and informatics; third, to provide a single forum within which commonalities and differences in the workings of complex systems may be discerned, hence leading to deeper insight and understanding. UCS will publish monographs, lecture notes and selected edited contributions aimed at communicating new findings to a large multidisciplinary audience.
- 8. Visarath In • Antonio Palacios Patrick Longhini Editors International Conference on Theory and Application in Nonlinear Dynamics (ICAND 2012) 123
- 9. Editors Visarath In Patrick Longhini Spawar Systems Center San Diego USA Antonio Palacios Department of Mathematics San Diego State University San Diego USA ISSN 1860-0832 ISSN 1860-0840 (electronic) ISBN 978-3-319-02924-5 ISBN 978-3-319-02925-2 (eBook) DOI 10.1007/978-3-319-02925-2 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2013955130 Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
- 10. Organizers Adi Bulsara, SPAWAR, San Diego Takashi Hikihara, Kyoto University, Japan Visarath In, SPAWAR, San Diego Ljupco Kocarev, Macedonain Acad. Patrick Longhini, SPAWAR, San Diego Antonio Palacios, San Diego State University Michael F. Shlesinger, Office of Naval Research Hiroyuki Torikai, Osaka University, Japan Mark Spano, Arizona State University Sponsor Office of Naval Research (ONR) 875 N. Randolph St., Suite 1475 Arlington, VA 22217 Army Research Office Engineering Sciences Directorate Mechanical Sciences Division 2800 Powder Mill Road Adelphi, MD 20783-1197
- 11. Preface Over the past three decades, the field of Nonlinear Science has evolved from being a valuable theoretical tool to study physical systems with dynamic behavior in space and time to a critical component to model, design and fabricate actual devices that exploit the inherently nonlinear features of many natural phenomena. Yet, while there has been significant progress in developing theoretical ideas and methods to study nonlinear phenomena under an assortment of system boundary conditions and preparations, there exist comparatively fewer devices that actually take advantage of the rich behavior exhibited by theoretical models. Consider, for instance, the fact that a shark’s sensitivity to electric fields is 400 times more powerful than the most sophisticated, currently available, electric field sensor. In fact, despite significant advances in material properties, in many cases it remains a daunting task to duplicate the superior signal processing capabilities of most animals. Bridging the gap between theory and biologically inspired devices can only be accomplished by bringing together researchers working in theoretical methods in nonlinear science with those performing experimental works. Other areas of strong interest among the research community, where theoretical findings can one day lead to novel technologies that exploit nonlinear behavior, include: chaos gates, social networks, communication, sensors, lasers, molecular motors, biomedical anomalies and stochastic resonance. A common theme among these and many other related areas is the fact that nonlinear systems tend to be highly sensitive to perturbations when they occur near the onset of a bifurcation. This behavior is universal among many nonlinear phenomena and, if properly understood and manipulated, it can lead to significant enhancements in systems response. Representative examples have been observed in a large number of laboratory experiments on systems ranging from solid state lasers to superconducting loops, and such behavior has been hypothesized to account for some of the more striking information-processing properties of biological neurons. Furthermore, background noise can precipitate this behavior, thereby playing a significant role in the optimization of the response of these systems to small external perturbations. Since 2005, we have held a series of meetings to bring together researchers across various disciplines working on theory and experiments in nonlinear science. The first meeting was 2005 Device Applications of Nonlinear Dynamics (DANOLD) meeting, held in Catania, Italy. Then in 2007 ICAND, the research vii
- 12. community met again in Poipu Beach, Koloa (Kauai), Hawaii, USA. More recently, the 2010 ICAND meeting was held in Alberta, Canada, at the luxurious Fairmont Chateau in Lake Louise. And, of course, 2012 ICAND was held in Seattle, Washington, USA. This last meeting brought together researchers from physics, engineering, and biology who were involved in the analysis and devel- opment of applications that incorporate and, indeed, exploit the nonlinear behavior of certain dynamical systems. The focus for 2012 ICAND was equally divided between theory and implementation of theoretical ideas into actual devices and systems. Contemporary topics on complex systems, such as social networks, were also featured among selected lecturers. The organizers extend their sincerest thanks to the principle sponsors of the meeting: Army Research Office (Washington, DC), Office of Naval Research (Washington, DC), Office of Naval Research-Global (Tokyo), San Diego State University (College of Sciences), and SPAWAR Systems Center Pacific. A special mention to Dr. Samuel Stanton from the Army Research Office and to Dr. Michael Shlesinger from the Office of Naval Research for their support and insight to hold such a diverse meeting. In addition, we extend our appreciation to Tania Gomez at SDSU for their hardwork in preparation and financial duty, which enabled the conference to run smoothly. We would also like to thank our colleagues who chaired the session and to all the personal who spent many hours making this meeting a success. Finally, we thank Springer for their production of an elegant proceeding. San Diego, USA, August 2013 Visarath In Patrick Longhini Antonio Palacios viii Preface
- 13. Contents Part I Invited Speakers Exploiting Chaos for Quantum Control . . . . . . . . . . . . . . . . . . . . . . . 3 Ying-Cheng Lai Random Vibration of a Nonlinear Autoparametric System . . . . . . . . . 11 Kristjan Onu, Nishanth Lingala and N. Sri Namachchivaya Nonlinear Dynamics of an Array of Nano Spin Transfer Oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 B. Subash, V. K. Chandrasekar and M. Lakshmanan Interactive Ensembles of Imperfect Models: Lorenz 96 System . . . . . . 39 Lasko Basnarkov and Ljupčo Kocarev Synthetic Computation: Chaos Computing, Logical Stochastic Resonance, and Adaptive Computing . . . . . . . . . . . . . . . . . . . . . . . . . 51 Behnam Kia, K. Murali, Mohammad-Reza Jahed Motlagh, Sudeshna Sinha and William L. Ditto Delay Differential Equation Models of Normal and Diseased Electrocardiograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Claudia Lainscsek and Terrence J. Sejnowski Voltage Response of Non-Uniform Arrays of Bi-SQUIDs . . . . . . . . . . 77 Patrick Longhini, Susan Berggren, Anna Leese de Escobar, Antonio Palacios, Sarah Rice, Benjamin Taylor, Visarath In, Oleg A. Mukhanov, Georgy Prokopenko, Martin Nisenoff, Edmond Wong and Marcio C. De Andrade A Bistable Microelectronic Circuit for Sensing Extremely Low Electric Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Andy Kho, Visarath In, Patrick Longhini, Daniel Leung, Norman Liu, Antonio Palacios, Joseph Neff and Adi Bulsara ix
- 14. Asynchronous Cellular Automaton Based Modeling of Nonlinear Dynamics of Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Hiroyuki Torikai and Takashi Matsubara Nonlinear Behaviour of Vital Physiological Systems . . . . . . . . . . . . . . 113 Eugenijus Kaniusas Fabrication of Bistable MEMS Systems for Energy Harvesting. . . . . . 123 Teresa Emery Symbolic Toolkit for Chaos Explorations . . . . . . . . . . . . . . . . . . . . . . 129 Tingli Xing, Jeremy Wojcik, Roberto Barrio and Andrey Shilnikov Enhancing Signal Resolution in a Noisy Nonlinear Sensor. . . . . . . . . . 141 Alexander P. Nikitin, Nigel G. Stocks and Adi R. Bulsara Distributed Bandpass Filtering and Signal Demodulation in Cortical Network Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Mark D. McDonnell Majority Rule in Nonlinear Opinion Dynamics. . . . . . . . . . . . . . . . . . 167 Michael Gabbay and Arindam K. Das Recent Advances on Information Transmission and Storage Assisted by Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 P. I. Fierens, G. A. Patterson, A. A. García and D. F. Grosz Transmission of Packets on a Hierarchical Network: Avalanches, Statistics and Explosive Percolation. . . . . . . . . . . . . . . . . 193 Neelima Gupte and Ajay Deep Kachhvah A Chaos MIMO-OFDM Scheme for Mobile Communication with Physical-Layer Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Eiji Okamoto Acoustic Detection and Ranging Using Solvable Chaos . . . . . . . . . . . . 213 Ned J. Corron, Mark T. Stahl, Jonathan N. Blakely and Shawn D. Pethel Using Phase Space Methods for Target Identification . . . . . . . . . . . . . 225 Thomas L. Carroll and Frederic J. Rachford Effect of Voltage Oscillations on Response Properties in a Model of Sensory Hair Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Rami M. Amro and Alexander B. Neiman x Contents
- 15. Low-Pass Filtering of Information in the Leaky Integrate-and-Fire Neuron Driven by White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Benjamin Lindner Application of High Performance Computing for Simulating the Unstable Dynamics of Dilute Spark-Ignited Combustion . . . . . . . . 259 Charles E. A. Finney, Miroslav K. Stoyanov, Sreekanth Pannala, C. Stuart Daw, Robert M. Wagner, K. Dean Edwards, Clayton G. Webster and Johney B. Green Investigating the Use of Manifold Embedding for Attractor Reconstruction from Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Lucas A. Overbey and Colin C. Olson The Dynamics of Coupled Spin-Torque Nano Oscillators: An Initial Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 J. Turtle, A. Palacios, V. In and P. Longhini Part II Posters Elaboration of a Multispecies Model of Solid Tumor Growth with Tumor-Host Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 A. Konstorum, S. A. Sprowl, A. D. Lander, M. L. Waterman and J. S. Lowengrub Resistive Switching Assisted by Noise. . . . . . . . . . . . . . . . . . . . . . . . . 305 G. A. Patterson, P. I. Fierens and D. F. Grosz Topology Independent SIS Process: Theory and Application. . . . . . . . 313 Igor Tomovski, Igor Trpevski and Ljupčo Kocarev The Role of Duty Cycle in a Three Cell Central Pattern Generator . . 319 Jeremy Wojcik, Robert Clewley and Andrey Shilnikov On the Approach of Ensemble of Interacting Imperfect Models . . . . . 327 Miroslav Mirchev and Ljupco Kocarev Synchronization in Coupled MEMS Resonators . . . . . . . . . . . . . . . . . 333 Suketu Naik and Takashi Hikihara Contents xi
- 17. Exploiting Chaos for Quantum Control Ying-Cheng Lai Abstract The field of Quantum Chaos is referred to as the study of quantum behaviors of systems whose corresponding classical dynamics are chaotic, or study of quantum manifestations of classical chaos. Equivalently, it means that quantum behaviors depend on the nature of the classical dynamics, implying that classical chaos can be used to control or manipulate quantum behaviors. We discuss two examples here: using transient chaos to control quantum transport in nanoscale sys- tems and exploiting chaos to regularize relativistic quantum tunneling dynamics in Dirac fermion and graphene systems. 1 Introduction Controlling chaos in dynamical systems has been studied for more than two decades since the seminal work of Ott, Grebogi, and Yorke [1]. The basic idea was that chaos, while signifying random or irregular behavior, should not be viewed as a nuisance in applications of nonlinear dynamical systems. In particular given a chaotic system, the fact that there are an infinite number of unstable periodic orbits embedded in the underlying chaotic invariant set means that there are an equally infinite number of choices for the operational state of the system depending on need, provided that any such state can be stabilized. Then, the intrinsically sensitive dependence on initial conditions, the hallmark of any chaotic system, implies that it is possible to apply small perturbations to stabilize the system about any desirable state. Controlling chaos has since been studied extensively and examples of successful experimental implementation abound in physical, chemical, biological, and engineering systems Y.-C. Lai (B) School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA e-mail: Ying-Cheng.Lai@asu.edu V. In et al. (eds.), International Conference on Theory and Application in Nonlinear 3 Dynamics (ICAND 2012), Understanding Complex Systems, DOI: 10.1007/978-3-319-02925-2_1, © Springer International Publishing Switzerland 2014
- 18. 4 Y.-C. Lai [2]. The vast literature on controlling chaos, however, has been limited to nonlinear dynamical systems in the classical domain. Recently, it has been articulated that chaos can be exploited to control or manip- ulate quantum-mechanical behaviors [3, 4]. For example, in the context of quantum transport through nanostructures, a fundamental characteristic is conductance fluc- tuations. It has been demonstrated that chaos, more specifically transient chaos, can be effective in modulating the conductance-fluctuation patterns, and it has been suggested [4] that this can be realized experimentally by applying an external gate voltage in a device of suitable geometry to generate classically inaccessible poten- tial barriers. Adjusting the gate voltage allows the characteristics of the dynamical invariant set responsible for transient chaos to be varied in a desirable manner which, in turn, can induce continuous changes in the statistical characteristics of the quantum conductance-fluctuation pattern. In another example, it has been shown that chaos can be utilized to suppress, significantly, the spread in the tunneling rate commonly seen in systems whose classical dynamics are regular, and this is called regularization of quantum tunneling dynamics by chaos [3]. More recently, it has been demonstrated that similar effects arise in relativistic quantum systems [5]. This Brief Review has two purposes: (1) to discuss the two aforementioned examples of exploiting chaos for quantum control, and (2) to argue that the prin- ciple of chaos-based quantum control is more general with potentially significant applications in nanoscience. 2 Controlling Quantum Transport by Transient Chaos A fundamental quantity characterizing the transport of an electron through a nanos- tructure, such as a quantum dot or a quantum point contact, is quantum transmission probability, or simply quantum transmission. In general, quantum transmission is determined by many electronic and system parameters such as the Fermi energy, the strength of external magnetic field (if there is one), and the details of the geometry of the structure. If the structure is connected through electron waveguides (or leads) to electron reservoirs (i.e., contacts) to form a circuitry, the conductances defined with respect to various voltage biases among the contacts, together with the correspond- ing currents, will be determined by the quantum transmission [6]. This means that the conductances can also depend sensitively on electronic and geometrical para- meters. For example, as the Fermi energy of the electron changes, the conductances can exhibit wild fluctuations and sharp resonances [7–13]. In applications such as the development of electronic circuits and nanoscale sensors, severe conductance fluctuations are undesirable and are to be eliminated so that stable device operation can be achieved. The outstanding question is then, can practical and experimentally feasible schemes be articulated to modulate quantum conductance fluctuations? It has been demonstrated recently and understood theoretically that classical transient chaos can be used to effectively modulate conductance-fluctuation patterns associ- ated with quantum transport through nanostructures [4].
- 19. Exploiting Chaos for Quantum Control 5 Intuitively, the basic principle underlying the transient-chaos based strategy for modulating quantum transport is quantum interference. It has been known that quan- tum pointer states, which are resonant states of finite but long lifetime formed inside the nanostructure [14–16], can cause sharp conductance fluctuations - a kind of Fano resonance [17, 18]. To give a specific example, consider quantum-dot systems, a par- adigm for investigating all kinds of quantum transport phenomena through nanostruc- tures. Such a system typically consists of a finite device region of certain geometrical shape, such as a square, a circle, or a stadium, and a number of leads connected with the device region. For a quantum-dot system whose classical dynamics is either regu- lar or contains a significant regular component, there are stable periodic orbits in the classical limit. If the dot geometry is closed, highly localized states can form around the classically stable periodic orbits as a result of quantum interference. When elec- tronic waveguides (leads) are attached to the quantum dot so that it is open, some periodic orbits can still survive, leading to resonant states, or quantum pointer states. Since the corresponding classical orbits are stable, the resonant states can have long lifetime, so their coupling to the leads is weak. As a result, narrow resonances can form around the energy values that are effectively the eigenenergies for the stable pe- riodic orbits in the corresponding closed system. When the dot geometry is modified so that the underlying classical dynamics becomes fully chaotic, no stable periodic orbits can exist. Although scars can still form around classically unstable periodic orbits in a closed chaotic system [19], the corresponding resonant states in the open system generally will have much shorter lifetimes. This means that these resonant states do couple to the leads more strongly, broadening the narrow resonances in the conductance-fluctuation pattern. Here chaos is transient because the system is open. According to the theory of transient chaos [20], the dynamical invariant sets responsible for transient chaos are non-attracting chaotic sets in the phase space. If the properties of transient chaos can be adjusted experimentally by parameter tuning, the quantum conductance fluctuation-patterns can then be controlled in a desirable manner. For example, one can change the effective geometry of the dot structure con- tinuously so as to enhance the escape rate, a basic quantity characterizing transient chaos, and this could lead to significantly smoother quantum-conductance fluctua- tions. To realize quantum control by using chaos, we conceive generating a region about the center of the device or structure with high potential so that it is impenetrable to classical particles. For example, consider a square quantum dot, a prototypical model in semiconductor two-dimensional electron-gas (2DEG) systems. When the dot is closed, the corresponding classical dynamics is integrable so that extremely nar- row resonances can arise in the quantum transport dynamics of the corresponding open-dot system. Now imagine applying a gate voltage perpendicular to the device plane to generate a circular, classically forbidden region about the center of the dot, as shown schematically in Fig.1. In general, the potential profile will be smooth in space. However, qualitatively, the classical scattering behavior is similar to that from an infinite potential well. Thus it is reasonable to impose the infinite potential-well assumption for the central region, which defines a “forbidden” region. Varying the voltage V0 can change the effective radius R of the forbidden region. Classically, the
- 20. 6 Y.-C. Lai R Left Lead Right Lead Fig. 1 Illustration of a possible scheme to control transport through a two-dimensional quantum-dot system. When semiconductor materials (e.g., silicon) are used, the system is the traditional two- dimensional electron-gas (2DEG) system described by the Schrödinger equation in non-relativistic quantum mechanics. If the material is graphene, in certain energy regime the system is described by the Dirac equation in relativistic quantum mechanics. By applying a suitable gate voltage per- pendicular to the device plane to generate a circular forbidden region at the center of the device, the resulting closed system is a Sinai billiard, whose classical dynamics is fully chaotic. Open quantum-dot system can be formed by attaching two leads to the billiard system, one on the left and another on the right side. The classical dynamics of the device can thus be characterized as chaotic scattering closed system is thus a Sinai billiard [21], which is fully chaotic, insofar as the radius of the central potential region R is not zero. When leads are connected to the device region so as to open the system, chaos becomes transient. The dynamical character- istics of the underlying chaotic invariant set can be adjusted in a continuous manner by increasing the radius R [22]. Quantum mechanically we thus expect to observe increasingly smooth variations in the conductance with, e.g., the Fermi energy, which has been demonstrated [4] using both semiconductor 2DEG and graphene [23–26] systems. Insights into why classical chaos can smooth out quantum conductance fluctua- tions can also be gained from the semiclassical theory of quantum chaotic scattering [27–29]. In particular, in the semiclassical regime, it was established by Blümel and Smilansky that the energy autocorrelation function of the quantum transmission fluctuation is proportional to the Fourier transform of the particle-decay law in the classical limit [27]. For fully developed chaotic transport through a quantum dot, the decay law is exponential with the rate κ. As a result, the quantum energy correlation function decays as a Lorentzian function with the width given by κ, where is the Planck’s contant. In the theory of transient chaos [20], κ is the escape rate associated with the underlying non-attracting chaotic set. As the radius of the central potential region is increased, κ also increases. The energy autocorrelation function then decays more slowly, signifying less fluctuations, i.e., less number of sharp resonances in the quantum transmission. This semiclassical argument suggests that the degree of quantum transmission fluctuations can be controlled by classical chaos. Extensive numerical support for the principle of transient-chaos based control of quantum transport and a detailed theoretical analysis can be found in Ref. [4, 30].
- 21. Exploiting Chaos for Quantum Control 7 3 Effect of Chaos on Quantum Tunneling The principle of regularization of quantum tunneling by chaos can be understood by considering the prototypical system in Fig.2, which consists of two symmetrical cavities connected by a one-dimensional potential barrier along the line of symmetry. When the classical dynamics in each cavity is integrable, for sufficiently large energy the tunneling rate can have many values in a wide interval. Choosing the geometry of the cavity such that the classical dynamics become chaotic can greatly enhance and regularize quantum tunneling. Heuristically, this can be understood, as follows. When the potential barrier is infinite, each cavity is a closed system with an infinite set of eigenenergies and eigenstates. Many eigenstates are concentrated on classical periodic orbits, forming quantum scars [19]. For a classically integrable cavity, some stable or marginally stable periodic orbits can persist when the potential barrier becomes finite so that each cavity system is effectively an open quantum system. Many surviving eigenstates correspond to classical periodic orbits whose trajecto- ries do not encounter the potential barrier, generating extremely low tunneling rate even when the energy is comparable with or larger than the height of the potential barrier. The eigenstates corresponding to classical orbits that interact with the poten- tial barrier, however, can lead to relatively strong tunneling. In a small energy interval the quantum tunneling rate can thus spread over a wide range. However, when the classical dynamics is chaotic, isolated orbits that do not interact with the potential barrier are far less likely and, consequently, the states associated with low tunneling rates disappear, effectively suppressing the spread in the tunneling rate. The idea of using chaos to regularize quantum tunneling dynamics was first conceived and demonstrated in non-relativistic quantum systems governed by the Schrödinger equation [3]. Recently, the question of whether chaos can regularize tunneling in relativistic quantum systems has been addressed [5], where the motion of massless Dirac fermions in the setting of resonant tunneling was investigated to facilitate comparison with the non-relativistic quantum case. In general, it is a chal- Fig. 2 A closed quantum system of arbitrary shape in two dimensions Potential barrier
- 22. 8 Y.-C. Lai lenging task to solve the Dirac equation in a confined geometry, due to the difficulties to incorporate zero-flux boundary conditions and to remove artificial, non-physical effects such as fermion doubling as a result of spatial discretization. A numerical scheme has then been developed [31] to overcome these difficulties, which is based on constructing a physically meaningful, Hermitian Hamiltonian. Extensive compu- tations have revealed unequivocally the existence of surviving eigenstates that lead to extremely low tunneling rates. As for the non-relativistic quantum case, making the cavities classically chaotic can greatly regularize the quantum tunneling dynam- ics. To explore the practical implications, resonant tunneling devices made entirely of graphene [23] have been studied [5], where the tunneling rates for different en- ergy values have been calculated. Qualitatively similar results have been obtained to those for massless Dirac fermions. One unique feature for both the Dirac fermion and graphene systems, which finds no counterpart in non-relativistic quantum tunneling devices, is the high tunneling rate in the regime where the particle energy is smaller than the height of the potential barrier. This is a manifestation of the Klein-tunneling phenomenon [32–34]. A theory has been developed [5] to explain the numerical findings, which is based on the concept of self energies and the complex energy spectrum of the non-Hermitian Hamiltonian for the “open” cavity. 4 General Thoughts on Chaos-Based Quantum Control The general principle of chaos-based control of quantum behaviors is motivated by the term quantum chaos, which does not mean that there can actually be chaos in quantum mechanical systems but is referred to as the study of quantum manifestations of systems whose dynamics in the classical limit exhibit chaos [35, 36]. The basic reason that chaos may be ruled out in quantum systems is that the fundamental governing equations, the Schrödinger equation or the Dirac equation, are linear. At the present, there is tremendous literature on quantum chaos, where various quantum manifestations of classically chaotic systems have been studied. The general result is that distinct classical dynamics, integrable or chaotic, can lead to characteristically different quantum behaviors. Furthermore, different types of chaotic behaviors can generate distinct quantum manifestations. From the point of view of control, all these suggest that quantum behaviors can be manipulated or harnessed for desirable applications by choosing distinct classical dynamical behaviors, in particular chaotic dynamics. The two examples discussed in this Brief Review, control of quantum transport and quantum tunneling, are based on building chaos into the system. In the transport problem that involves the Sinai billiard type of device structure, the properties of the underlying chaotic set can be modified, for example, by an externally adjustable gate voltage. In the tunneling problem, the geometry of the cavities are deliberately designed to yield chaotic dynamics in the classical limit. Once the structure is fixed, experimentally it may be difficult to change the characteristics of chaos. It is thus nec-
- 23. Exploiting Chaos for Quantum Control 9 Fig. 3 A four terminal device in the presence of magnetic field. The area enclosed by the greenlines indicatesthe region in the physical space where transient chaos can occur. The black circle marks the region in which the magnetic field is applied essary to search for experimentally feasible schemes to modulate the characteristics of the underlying chaotic invariant sets in a continuous fashion. Figure3 presents a possible scheme where a single external parameter can be var- ied to realize chaos-based control of quantum transport. It is a four-terminal device, where four idealized leads join smoothly to form a quantum-dot structure, which has been used widely in the study of, for example, quantum Hall effect [6]. The struc- ture typically exhibits chaotic scattering (transient chaos) in the classical limit. A perpendicular magnetic field can be applied. An earlier work [37] demonstrated that the dynamical invariants of the underlying non-attracting chaotic set can be modi- fied continuously by changing the strength of the magnetic field. It is thus possible to modulate the quantum conductance-fluctuation patterns by simply adjusting the magnetic-field strength [38]. Acknowledgments The main idea of using chaos to manipulate quantum behaviors was generated through extensive discussions with Dr. L. Pecora from Naval Research Laboratory in January 2011 at Dr. M. Shlesinger’s ONR Program Review Meeting at UCSD. The computations and theoretical analyses reported in the references [4, 5, 30, 31] on which this Review is based were mainly carried out by Dr. R. Yang, Dr. X. Ni, and Dr. L. Huang, all formerly affiliated with ASU. References 1. E. Ott, C. Grebogi, J.A. Yorke, Phys. Rev. Lett. 64, 1196 (1990) 2. S. Boccaletti, C. Grebogi, Y.-C. Lai, H. Mancini, D. Maza, Phys. Rep. 329, 103 (2000) 3. L.M. Pecora, H. Lee, D.-H. Wu, T. Antonsen, M.-J. Lee, E. Ott, Phys. Rev. E 83, 065201 (2011) 4. R. Yang, L. Huang, Y.-C. Lai, L.M. Pecora, Appl. Phys. Lett. 100, 093105 (2012) 5. X. Ni, L. Huang, Y.-C. Lai, L.M. Pecora, EPL 98, 50007 (2012)
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- 25. Random Vibration of a Nonlinear Autoparametric System Kristjan Onu, Nishanth Lingala and N. Sri Namachchivaya Abstract We examine a stochastically forced autoparametric system for its stationary motion and stability. The deterministic form of this system is nearly Hamiltonian (with small dissipation) and exhibits 1:2 resonance and phase-locking. We develop a stochastic averaging technique to achieve a lower dimensional descrip- tion of the dynamics of this system. Stochastic averaging is possible due to three time scales involved in this problem. Each time scale is fully exploited while aver- aging. The dimensional reduction techniques developed here consist of a sequence of averaging procedures that are uniquely adapted to study stochastic autoparamet- ric systems. What motivates our analysis is that classical averaging methods fail when the original Hamiltonian system has resonances, because, at these resonances, singularities arise in the lower-dimensional description. At these singularities we introduce gluing conditions; these complete the specification of the dynamics of the reduced model. Examination of the reduced Markov process (which takes values on a nonstandard space) yields important results for probability density functions. 1 Introduction We investigate the random vibrations of a nonlinear autoparametric system of the form q̈1(t) + ζ1q̇1(t) + f1(q1(t), q2(t)) = ξ(t) q̈2(t) + ζ2q̇2(t) + f2(q1(t), q2(t)) = 0 t ≥ 0, (1) K. Onu (B) Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, QCH3A 2K6, Canada e-mail: kristjan.onu@mcgill.ca N. Lingala · N. Sri Namachchivaya Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA e-mail: lingala1@illinois.edu N. Sri Namachchivaya e-mail: navam@illinois.edu V. In et al. (eds.), International Conference on Theory and Application in Nonlinear 11 Dynamics (ICAND 2012), Understanding Complex Systems, DOI: 10.1007/978-3-319-02925-2_2, © Springer International Publishing Switzerland 2014
- 26. 12 K. Onu et al. where for each time t 0, (q1(t), q2(t)) represents the generalized coordinates of the system, the constants ζ1 and ζ2 are damping coefficients, and ξ(t) is a stationary random process. We are interested in questions of stability of the stochastic sys- tem (1), and in the transfer of energy from the forced mode q1 to the unforced mode q2. It is well known that, in the presence of 1 : 2 resonance and periodic excitation, as the intensity of excitation is increased, the excited mode reaches a certain value of amplitude at which saturation takes place and then the energy is transferred to the unforced mode. This may be undesirable, because disturbances affecting one mode may cause unwanted instability in another mode. Our effort is to answer whether the saturation and energy transfer occurs in the presence of noisy input. Towards this goal, we achieve a lower dimensional description of the above system. The dissipation and random perturbations are assumed to be small. This means that their effect will be visible only over a long time horizon. When the nonlinearities are also assumed small, the dominant part of the dynamics is that of two uncoupled oscillators. In particular, the dynamics of the unperturbed system identify a reduced phase space (the orbit space) on which to carry out stochastic averaging. While the classical theory of stochastic averaging is a natural framework for such a program, the equations of interest contain resonances and bifurcations, which precludes a simple application of classical techniques. In particular, the resonance gives rise to an intermediate scale, and the bifurcations give rise to some non-standard singularities in the orbit space. The equations of motion considered (1) can model the dynamics of a number of mechanical systems, namely a random excitation of a initially deformed shallow arch, a suspended elastic cable driven by planar excitation, or a water vessel subject to longitudinal wave action. To keep things as simple as possible, we shall consider a very simple system, namely a pendulum hanged from a mass which is attached by a spring to a support (Fig.1). The mass is randomly excited. For clarity, we use mass to refer to the object at the free end of the spring, while the object at the end of the pendulum is referred to as the bob. The quantity ϕ is the angle of the pendulum (with respect to the vertical axis) and the quantity y represents the height of the mass (relative to a rest position defined by the position of the pendulum). The mass is forced according to a stochastic signal Ξ(t). The subscripts here refer to the fact that this is our original physical model. The equations for such a system can be written as Fig. 1 Schematic of autopara- metric system o y f k d p
- 27. Random Vibration of a Nonlinear Autoparametric System 13 (mo + mp)ÿ + do ẏ + ky + mpl(ϕ̈ sin ϕ + ϕ̇2 cos ϕ) = Ξ, mpl2 ϕ̈ + dpϕ̇ + mpl(g + ÿ) sin ϕ = 0, (2) where mo, do and k are the mass, damping and the spring constant of the spring-mass system and mp, dp and l are the mass, damping and the length of the pendulum. The kinetic and the potential energies of the system are given by T = 1 2 (mo + mp)ẏ2 + 1 2 mp l2 ϕ̇2 + mp lẏ ϕ̇ sin ϕ, U = mp gl(1 − cos ϕ) + 1 2 ky2 . It is clear that the nonlinearities in the equations of motion arise due to the grav- itational restoring force and due to the dependence of kinetic energy on the angle ϕ which leads to inertial coupling between the the two coordinates. It also turns out (we shall use this later) that in the absence of noise and damping, this system is Hamiltonian, so the dynamics of y and ϕ are governed by the geometry of this Hamiltonian. The above equations in dimensionless coordinates are ¨ η̂ + 2ζ̂o ˙ η̂ + η̂ + R( ˙ θ̂ sin θ̂ + ˙ θ̂ 2 cos θ̂) = ξ̂(t), R ¨ θ̂ + 2Rζ̂p ˙ θ̂ + R(q2 sin θ̂ + ¨ η̂ sin θ̂) = 0. where Fig. 2 Surface and contour plots of K(u1, u2). I = 1
- 28. 14 K. Onu et al. Fig. 3 State space M for Yε t −200 −100 0 100 200 80 90 100 110 120 5 10 15 x 10 −5 H I Fig. 4 Probability density by FEM ω2 def = g l , ω2 o def = k mo + mp , q def = ω ωo , R def = mp mo + mp , ζ̂o def = do 2 k(mo + mp ) , ζ̂p def = dp (mo + mp ) 2l2 √ k = dp 2l2ωo . and where ξ̂(t) = ξ̂(t/ωo) kl , η̂(t) = y(t/ωo) l , θ̂(t) = ϕ(t/ωo) for all t 0. Our interest here is a refined stability analysis near the fixed point (η̂, θ̂) ≡ 0 of the unperturbed system. In particular, we are interested in the effect of small random perturbations, so we will let ξ̂ be of the form ξ̂ = ε2νξ, where ξ is a noise process of “unit” variance and ν is some empirical parameter. Our dynamics are most interesting when they are not over-damped, so let ζ̂o and ζ̂p be of the form ζ̂o = ε2ζo and ζp = ε2ζp, where ζo and ζp are some positive constants (this corresponds to letting do and dp be of size ε). Guided by the corresponding analysis for periodic
- 29. Random Vibration of a Nonlinear Autoparametric System 15 Fig. 5 Probability density by numerical simulation forcing, we are interested in the behavior when q2 is very close to q2 o ≡ 1/4. Let’s replace q by qo +ε2μ, where μ is an unfolding parameter. Since we are interested in η̂ and θ̂ near the fixed point 0, we should look at these quantities on a finer resolution. Namely, let η and θ be defined by η̂(t) = εη(t), θ̂(t) = εθ(t) then the dynamics of the system yields η̈ + 2ε2 ζoη̇ + η + R(θ̈ sin(εθ) + εθ̇2 cos(εθ)) = ενξ(t), Rθ̈ + 2ε2 Rζpθ̇ + R q◦ + ε2 μ sin(εθ) ε + η̈ sin(εθ) = 0, (3) where ε is a small scaling parameter, q0 = 1/2 signifying 1 : 2 resonance, μ is the parameter representing unfolding from the resonance, R is the ratio of mass of the unforced mode to the total mass. 2 Single Mode Solutions To clarify some general qualitative effects of noise, let’s consider a simple stability analysis using some spectral methods and the first-order linearization. The mass on the spring can move only in the vertical (η) direction and is excited by νξ. Assume that the pendulum is locked vertically, i.e. θ(t) ≡ 0. We get the equation η̈ + 2ε2 ζoη̇ + η = ενξ.
- 30. 16 K. Onu et al. If ξ is white noise we can solve for η explicitly. Its power spectral density is Sη(ω) = ε2ν2S0 (1 − ω2)2 + 4ε4ζ2 o ω2 , where S0 is the power spectral density of ξ. The peak intensity and the carrying frequency of η are determined by the filter parameter ζo. The stability of the locked mass steady-state oscillation is now obtained by using the first-order approximation of sine and cosine in the dynamics for θ. We get θ̈ + 2ε2 ζpθ̇ + ((q0 + ε2 μ)2 + εη̈)θ = 0, and the power spectral density of η̈ is given by Sη̈(ω) = ω4ε2ν2S0 (1 − ω2)2 + 4ε4ζ2 o ω2 . The maximal Lyapunov exponent can now be easily calculated and the stability boundary can be obtained in terms of excitation intensity ν and the dissipation coef- ficients ζp. An explicit expression for the maximal Lyapunov exponents of the single mode solution is given by expanding it in ε, we have λ1 ≈ ε2 −ζp + 1 8 q2 o Sη̈(2 (qo + ε2 μ)) and λ2 = ε2 −ζp − 1 8 q2 o Sη̈(2 (qo + ε2 μ)) . The noise has no effect on the other two exponents; i.e., λ3 = λ4 = −ε2ζo. Since the point θ ≡ 0 is a stable point for the hanging pendulum, the pendulum undergoes small random motion near θ ≡ 0, and all four Lyapunov exponents are negative. However, as we further increase the noise intensity, the top exponent becomes positive when ν2S0 = 8ζ2 o ζp. The system then becomes unstable, and the following question arises. • Do both the mass spring oscillator and the pendulum undergo random vibrations when the top exponent becomes positive (i.e., ν2 S0 8 ζ2 o ζp), i.e., does a new coupled-mode “stationary solution” or “stationary density function” appear? 3 Coupled Mode Solutions Making use of a time-varying symplectic transformation (see [1] for details), we arrive at ẋε t = εb1 (xε t , t) + ε2 b2 (xε t , t : ζ, μ) + εσ(xε t , t : ν)ξ(t), (4) where(x1, x2)and(x3, x4)areconjugatepairsandcanbethoughtofastheamplitudes of periodic orbits of the dominant dynamics.
- 31. Random Vibration of a Nonlinear Autoparametric System 17 The coefficients b1, b2, σ are periodic in time. Standard deterministic averaging can be used to average out the effects of rapidly-oscillating periodic coefficients. Let M be this averaging operator. Definition 1 (Time averaging operator) For a function ϕ ∈ C∞(R4 × R) which is 2π periodic in its last argument, define the time averaging operator M by (Mϕ)(x) ≡ 1 2π 2π 0 ϕ(x, t)dt. From the explicit formulas (see [1]) for b1 (where q = 1/2), we see that for x = (x1, x2, x3, x4) ∈ R4, (Mb1 )(x) = −1 2 x2x4, 1 2 (x1x4 − x2x3), 1 4 (x2 2 − x2 4 ), 1 2 (x1x2 + x3x4) . Then the averaged system ẋt = (Mb1)(x) is a 4-dimensional Hamiltonian system with two first integrals K and I in involution. The Hamiltonian associated with these dynamics is K(x) = 1 4 x1(x2 4 − x2 2 ) − 1 2 x2x3x4. (5) The unperturbed four-dimensional Hamiltonian system ż = ¯ ∇K(z) (6) has two first integrals in involution, namely, the Hamiltonian itself (5) and a second constant of motion (momentum variable) I (x) = (x2 1 + x2 3 ) + 1 2 (x2 2 + x2 4 ). (7) The invariant I is functionally independent of K, exists globally and is single valued. Note that the Hamiltonian system’s equations remain unchanged when t → −t, x1 → −x1 and x3 → −x3. 3.1 Dimensional Reduction Our main analytical tool is a certain method of dimensional reduction of nonlinear systems with symmetries and small noise. As the noise becomes asymptotically small, one can exploit symmetries and a separation of scales to use well-known methods (viz. stochastic averaging) to find an appropriate lower-dimensional description of the system.
- 32. 18 K. Onu et al. In the flow given by (4), the quantities (K(x), I (x)) are slow-varying. The variation of yε t := (K(xε t ), I (xε t )) is given by the following set of equations ẏε t = εF1 (xε t , t) + ε2 F2 (xε t , t : ζ, μ) + εG(xε t , t : ν)ξ(t), (8) where Fi j (x, t) = (bi (x, t).∇)yj and G j (x, t) = (g(x, t).∇)yj . Since K and I are integrals of motion for ẋt = (Mb1)(x), it is clear that MF1(X) = 0. Thus, to see the fluctuations of K and I, we need to look on a time scale of order 1/ε2. Thus, we make a time rescaling, setting Xε t def = xε t/ε2 and Yε def = yε t/ε2 . Then we have Ẋε t = 1 ε b1 (Xε t , t/ε2 ) + b2 (Xε t , t/ε2 ) + g(Xε t , t/ε2 ) 1 ε ξ(t/ε2 ), Ẏε t = 1 ε F1 (Xε t , t/ε2 ) + F2 (Xε t , t/ε2 ) + G(Xε t , t/ε2 ) 1 ε ξ(t/ε2 ). (9) Roughly, our goal is to study (9) and show that as ε tends to zero, the dynamics of Yε(Xε t ) tends to a lower-dimensional Markov process and to identify the infinitesimal generator of the limiting law. There are three time scales. The periodic fluctuations of the coefficients occur over time scales of order ε2. The effects of drift due to b1 can be seen on time scales of order ε. The drift and diffusion coefficients of Yε t are of order 1. We perform two averaging steps, one to average (M) the periodic behavior of the coefficients, and one to average (A) along the orbits of the Hamiltonian system ẋt = (Mb1)(x). To understand the state space of the slow variable Yε t , we consider the following symplectic transformation (it would also be useful later for simplifying calculations). 3.1.1 Structure of the Unperturbed System: Hamiltonian Structure x1 = u1 cos(2ψ) + u2 sin(2ψ), x3 = −u1 sin(2ψ) + u2 cos(2ψ), x2 = 2(I − u2 1 − u2 2) sin ψ, x4 = 2(I − u2 1 − u2 2) cos ψ. (10) The conjugate pairs are (u1, u2) and (ψ, I). This transformation yields u̇1t = −u1t u2t , u̇2t = 1 2 (3u1 2 t + u2 2 t − It ), ψ̇t = 1 2 u1t , ˙ It = 0 (11) and the corresponding Hamiltonian is K = 1 2 u1 I − (u2 1 + u2 2) . (12)
- 33. Random Vibration of a Nonlinear Autoparametric System 19 The relation between K, u1, u2, and I is illustrated in Fig.2. Note that this system’s equations remain unchanged when t → −t, u2 → −u2 and ψ → −ψ. System (11) has four fixed points. They are (u1, u2) = (0, ± √ I) and (u1, u2) = (± √ 3I 3 , 0). The points on the u1 axis are saddle points and those on the u2 axis are center fixed points. 3.1.2 State Space of Yε t The slow variable Yε t evolves on an arrowhead. Let S def ={x ∈ R4 : K∗ K(x) K∗, 0 I (x) I∗}. Then define an equivalence relation ∼ on R4 by identifying x ∼ y if x and y are on the same orbit of the hamiltonian flow ẋt = (Mb1)(x). Define M def = S̄/ ∼, and endow M with the quotient topology defined by ∼. If x ∈ S̄, we let [x] := {y ∈ S̄ : y ∼ x} be the equivalence class of x. π(x) := [x]. The slow variable Yε t evolves on M = 2 i=1 Γi ∪ 2 i=0[ci ] ∪ 2 i=1 i where ci are the fixed points, the i are closed orbits whose union is ∂S̄, and each Γi is the π-image of a maximal open subset of R4 which does not intersect any of the [ci ] or i . The state space is illustrated in Fig.3. 3.1.3 M A Averaging If the external noise ξ(τ) represents mean zero, stationary, independent stochas- tic processes with the strong mixing property, then roughly, as ε → 0, 1 ε ξ(t/ε2) approaches a white noise process. Khasminskii [2] gave a rigorous proof that a fam- ily of processes Xε t converges to a diffusion process. The aim here is to make use of this and derive a reduced graph-valued process for the integrals of motion, Yε. We have pointed out that that there are three time-scales involved in our averaging problems. The first step is to average the periodic fluctuations of the coefficients and obtain M-averaged quantities as the precursors to the stochastically averaged drift and diffusion coefficients. Somewhat laborious calculations yield the M-averaged quantities mi (x) ≡ M F2 1 + f1 + g1 (x) and ai j (x) ≡ M σσT i j (x). (13) These calculations can be simplified by considering the symplectic transformation (10) which provides a convenient geometric structure of the unperturbed integrable Hamiltonian problem. In (K, I, u) coordinates, the drift and diffusion (13) coeffi- cients are m1(K, I, u) = −(ζo + 2ζp)K − 1 4 (8μ + 3I) K u2 u1 + 1 2 3 + 1 R K2 u2 u2 1 , m2(K, I, u) = 2[σ2 Sξξ (1) − ζo I + 2(ζo − ζp)K/u1], (14)
- 34. Another Random Document on Scribd Without Any Related Topics
- 35. Telegrams Cables Longing, London Telephone No. 9313 Central June—Sept. 1907 MR. JOHN LONG'S SUMMER AND EARLY AUTUMN BOOKS 1907 SIX SHILLING NOVELS Crown 8vo., cloth gilt ITINERANT DAUGHTERS By Dorothea Gerard (Mme. Longard de Longgarde) In this story Dorothea Gerard tells of the result of an experiment, invented and tried on four girls, to whom home life had become irksome, by an up-to-date doctor. This experiment forms, in the main, the peg on which the story hangs, and which dissects and lays bare the characters of the heroines. The manner in which the girls meet the fate falling to them provides a great variety from the ordinary love-story, and the novel promises to be one of the most original and amusing published for some considerable time. ONLY BETTY By Curtis Yorke. With Coloured Frontispiece by E. J. Sherie In Only Betty Curtis Yorke has imagined a story which gives full rein for the display of those gifts of tenderness, naturalness, and distinction which readers and critics alike associate with her work. Only Betty is one of a large family left in poverty by the death of their father, and she answers an advertisement for services in a remote Welsh village. Betty obtains the post, and the authoress proceeds with great vivacity and charm to describe the lively series of events which follow. Curtis Yorke's popularity grows with every new book she produces, and her public will be immeasurably increased by her latest. MRS. BARRINGTON'S ATONEMENT By Violet Tweedale
- 36. In Violet Tweedale's new novel an excellent plot is unfolded with subtlety and force. It would spoil the reader's pleasure to enter fully into details—the curious psychic experiences, the tragedy and pathos of an immature soul, misunderstanding and misunderstood—but we can promise to those who read the novel that they will not find a dull page in this newest work of a writer to whom we can always look for novelty, brilliance and substantial interest. THE WHITE HAND AND THE BLACK By Bertram Mitford Mr. Bertram Mitford has done for South Africa what Mr. Rudyard Kipling has done for India. He has brought home to the English people the character of the work that Britons are doing in the outposts of Empire. Mr. Mitford's knowledge, like Mr. Kipling's, has been acquired at first hand, by living in the land and among the people he describes. In his new novel the author chooses as background a Rising of the Blacks against the Whites. The reader is brought into contact with various kinds of natives, good and bad, with the British official of the better class, and with the grit and solidity and daring of the ordinary Britisher who finds himself in a tight corner and fights with his back against the wall. Trickling through the stirring incidents of the story is a love romance. Mr. Mitford has intimate knowledge, insight, sympathy and imagination, and he has written a novel of virility and vigour whose superiority to most fiction may be observed on every page. DELILAH OF THE SNOWS By Harold Bindloss No living writer has a more intimate knowledge of colonial manners than Mr. Harold Bindloss. He describes for the stay-at-home Englishman not so much the well-ordered life in the great settlements as the virile, rugged, desperate, and often lawless struggles among the colonists in the undeveloped outposts of Empire. The earlier scenes in Delilah of the Snows take place in England. Later on the characters are transplanted bodily to Western Canada among the gold- seekers. In such surroundings Mr. Harold Bindloss, as may be conjectured, is in his element, and he develops a story of consummate artistry and strength. The spirit of adventure and tragedy and comedy is over it all, and an unconventional ending is in keeping with the rest of this brilliant book. DR. MANTON By Morice Gerard Mr. Morice Gerard has advanced with rapid strides to the position of one of the most popular writers of the day. Dr. Manton is a splendid instance of his power in weaving a dramatic story, made up of the great elements of love, mystery and conflict between opposing forces, with a wonderful dénouement, which no reader can read without being moved. Mr. Morice Gerard believes in a happy ending; hero
- 37. and heroine find their happiness and peace achieved after stress and struggle. The story is up to date in every respect. A WOMAN PERFECTED By Richard Marsh A new story by Mr. Richard Marsh is an event which is eagerly anticipated; and A Woman Perfected will not disappoint the expectations, however high, of any of Mr. Marsh's innumerable admirers. The starting-point of the story is the sudden death of a man of mysterious habits and ostentatious wealth, whose only daughter, Nora, is apparently left unprovided for. The young girl has been led to believe that she would be a great heiress, but the secret of her father's past and the source of his income cannot be discovered. A series of events follow, which excite a curiosity that amounts to anxiety. The author marshals his plot and characters with conscious mastery; and he has written what may, with very truth, be described as a brilliant book. CYNTHIA IN THE WILDERNESS By Hubert Wales. Author of Mr. and Mrs. Villiers, The Yoke This book almost reverses the question raised by the author's earlier work, Mr. and Mrs. Villiers. It is a study of a wife who, through the incapacity of her husband to understand or respond to the deeper woman in her, finds herself shut out upon the wilderness of joyless things. Mr. Hubert Wales has made his mark as an author, and his first two books, Mr. and Mrs. Villiers and The Yoke, have been out-standing successes. HER FATHER'S SOUL By Lucas Cleeve An incident which occurs in India between a Native Prince and an English Peeress is the source whence the subsequent events spring. The power of the story lies in its imagination and its phantasy. Lucas Cleeve has the great gift of expression, and in Her Father's Soul, she enables the reader to realize something of the weird, mysterious beauty and fascination of the land of the Oriental. THE WHITE COUNTESS By Florence Warden It may safely be stated of Miss Florence Warden's new story, The White Countess, that before the reader has reached the end of the first chapter he will find himself immersed in a mystery of baffling complexity, and that the sensational events which follow in swift succession will give him no pause until the last lines are in sight. The White Countess is a story of action and plot, and it will uphold
- 38. Miss Florence Warden's reputation as a writer of straightforward, dramatic, and exciting fiction. THE CURSE OF THE FEVERALS By L. T. Meade. With coloured Frontispiece by E. J. Sherie Mrs. L. T. Meade has chosen the subject of heredity as the theme of her new novel; but, as might be imagined, there is nothing unpleasant or technical in her treatment of The Curse of the Feverals. On the contrary, Mrs. Meade invariably looks at the brighter side of life—upon its joys rather than its sorrows—and she has brought her best talents to bear in the construction of this effective and moving story of domestic life. THE HOUSE ON THE THAMES By G. W. Appleton A boating accident on the Thames, the rescue of Jeannie, a beautiful girl, and her sudden disappearance almost immediately afterwards, are the events which occur in the first chapter of Mr. G. W. Appleton's new novel. Thenceforward the reader's perplexity is mingled with an intense desire to probe the mystery. Mr. Appleton keeps well within the region of probability, and his sunny outlook upon life peeps forth in this exciting, dramatic, and withal humorous story. KINDRED SPIRITS By L. T. Meade Deals with the passionate love of two girls for one man, and shows how the one whose love he did not return yet loved him so thoroughly, so nobly, so unselfishly, that in the end she was the means of his salvation. The story largely deals with an old family curse, and a strange mystery which is partially founded on fact. The characters of some of the most important persons are taken from life. In a word, this is the most exciting story that Mrs. Meade has ever produced, and the publisher predicts a more than ordinary success for it. IN HIS GRIP By David Christie Murray Mr. David Christie Murray has imagined in his new story a combination of circumstances which afford ample scope for the exercise of his uncommon powers. A merchant of character is left, by a dying friend, in the position of trustee without documentary conditions, and the property, which he thinks to be worthless, proves to be of untold value. His own financial embarrassments create the temptation to which he momentarily succumbs. The story gallops along at a furious pace amid an atmosphere of stirring events, through which runs a delightful love episode.
- 39. A JACOBITE ADMIRAL By R. H. Forster In the opinion of the critics, Mr. R. H. Forster knows Northumbria as Mr. Hardy knows Wessex, as Mr. Crockett knows Galloway, and as Mr. Blackmore knew Exmoor. Higher praise for a writer of historical fiction it would be difficult to imagine. In A Jacobite Admiral the Jacobite rebellion of 1715 forms the superstructure of the story, and the hero's adventures in his loyalty to the doomed house of Stuart, as conspirator, rebel, fugitive, and lover, afford ample scope for the description of scenes and localities of great natural beauty and historic interest. Over and above this, Mr. R. H. Forster has written a novel which is instinct with the finest romantic spirit. IN THE QUEEN'S SERVICE By Dick Donovan Dick Donovan has gone back to the troublous times of Mary Queen of Scots for the incidents which form the basis of his new romance, In the Queen's Service. Manners and morals were doubtless less refined and less humane in those days, and objects were pursued with more violence and more disregard of consequences. From the novelist's standpoint, the period is rich in materials and possibilities. Intrigue, treachery, murder, disaster, chivalry, gallantry, passion, self- sacrifice—these are the constituents of In the Queen's Service, and the author has, with his accustomed skill, created from them a story of great and penetrating interest. PURPLE HEATHER By Alan St. Aubyn The late Hawley Smart was a master of the true sporting novel, and Alan St. Aubyn has run him very close in Purple Heather. In this she has given us a picture of the wilds of Exmoor the whole year round, with vivid descriptions of some of the inhabitants, and at the same time has woven about them a story of very human interest which centres on the packs of stag and fox hounds. THE SECRET SYNDICATE By Fred Whishaw To the English reader there is a perennial fascination in the conditions of life in the mighty realms of the Czar, and few English writers have a more intimate knowledge of the various revolutionary currents in Russian affairs than Mr. Fred Whishaw. The action of The Victims transpires mainly in St. Petersburg, and the characters are wholly Russian. A young journalist and a young girl of the landed class are the central figures, and around these two are described the events which culminate in the recent peasant revolt. The story is graphically told, and has the air of being a veritable transcript from life.
- 40. THE PLEASURE MONGER By C. Ranger-Gull The novel which tells of a complete, triumphant, and overwhelming success is always delightful to read. This is the case in Mr. Ranger Gull's new book, The Pleasure Monger, one which will be found to be the best of all the brilliant studies of modern life which this author has given us. The character studies are especially strong and vivid, and the keen love-interest, which runs like a scarlet thread through the warp and woof of the tale, is novel and daring. It is very rarely, moreover, that a well-known author lifts the veil of the modern literary life and shows it as it really is. Stories which deal in part with literary life are nearly always written by amateurs. The Pleasure Monger, in short, will be found full of force, brilliancy, and interest. THE SIN OF GABRIELLE By Mrs. Coulson Kernahan In this story Mrs. Kernahan has shown in the character of Gabrielle Desturnelle a beautiful young French adventuress, who has all the seductive charm of Fanchette, without that heroine's innocence. The story shows how, by her arts and unscrupulous cleverness, she wrecks the life of Donovan Fitzgerald, a man of high ideals. The heartlessness of Gabrielle is put forth with power, while the nobility of Fitzgerald will win the sympathy of the reader. THE SECOND BEST By Coralie Stanton Heath Hosken The work of Coralie Stanton and Heath Hosken stands out from the fiction of to- day in bold outline; there is nothing commonplace or anæmic about it. Their new novel, The Second Best, is a realistic story of modern English society; the characters are vivid and natural, and the incidents palpitate with drama. The title is in keeping with the underlying idea, but The Second Best is, in point of fact, the authors' very best; it grips from the first, and a rich treat is in store for those who come within the spell of this rousing story. A LOVELY LITTLE RADICAL By Alice M. Diehl The work of Alice M. Diehl has two main recommendations—its freedom from the unpleasant and its polished and artistic setting. A Lovely Little Radical, however, is not a placid story. It recounts the love of a young girl of patrician birth and heritage for a simple man of the people. The author manipulates her theme with unfailing tact and discrimination, and succeeds in eliciting the reader's sympathies from the commencement. A Lovely Little Radical may be regarded as the crowning achievement of this popular romanticist.
- 41. THE LADY OF THE BLUE MOTOR By G. Sidney Paternoster. With Coloured Frontispiece Mr. Paternoster seems determined to prove in this novel that the motor-car provides inimitable opportunities for the rebirth of romance in an unromantic twentieth century. The central character, My Lady Melodé, is a veritable heroine of romance. From the time she makes her appearance, enshrouded in a cloud of mystery, in the salon of a hotel at Versailles, she is the centre of a series of thrilling adventures and dramatic situations which enchain the reader's attention until the happy outcome is reached. INNOCENT MASQUERADERS By Sarah Tytler This story is founded on an incident which happened at Blackheath upwards of thirty years ago. Two baby girls were exposed and left on the same night at opposite points of the heath to the charity of wayfarers. There is abundance of interest and incident before the mystery of their origin is solved. On one occasion the wrong waif is installed with an old city knight and his lady as their lost grandchild and heiress, but all comes right in the end. Best of all, the two waifs are innocent, good girls, although their adventures should appeal to every lover of true romance. RUBINA By James Blyth. With Coloured Frontispiece Mr. James Blyth stands almost alone among English novelists as a realist of the Zola School, but superadded to his realism is a strain of thought at once subtle and poetical. Rubina is the story of a girl of the people, and her life is passed wholly in a village in the heart of the Fens. Surrounded by the sights and sounds of nature, she herself is a child of nature, untrammelled by the niceties and scruples of modern conventions. Mr. Blyth pursues his theme in a legitimate and logical fashion, and he has produced a work which is a veritable piece of life, the poignant emotional power and truth of which will be acknowledged by every thoughtful reader. THE PASSING OF NIGHT By J. Fovargue Bradley The Passing of Night is a political novel written with a purpose. The author is a Congregational Minister who does not write at random, but whose views will be found worthy of attention, if they do not find acceptance. The story, from its argumentative side, treats of the attitude of the Church Association towards the Ritual excesses in the Church of England, and of Disestablishment in the interests
- 42. of religious and social life; but the author challenges the advocates of Disendowment to show the equity of their case. The Passing of Night is polemical and controversial, but it is also a romance of consummate interest; there is wit, imagination, insight, sense of character, and high literary quality in it. It is a first work, but it is certain to be regarded as one of the most remarkable novels of the year. A BRIAR ROSE By Sarah Tytler It is a characteristic of most fiction that the last chapters close to the sound of wedding bells. In A Briar Rose the order has been reversed, and the marriages take place at the commencement of the story. Miss Sarah Tytler has chosen the everyday lives of two young couples as her foundation, and around their joys and sorrows she has written a domestic story of quiet and penetrating charm. In this book, as in all her works, Miss Sarah Tytler's delicate literary gifts are distinctively apparent. LITTLE JOSEPHINE By L. T. Meade. With Coloured Frontispiece by E. J. Sherie There is scarcely a household in which the novels of Mrs. L. T. Meade are not known and appreciated; her work is infinite in its variety, and never dull. The thesis of her new story, Little Josephine, is the marriage of a good and charming young girl with a man of blemished character. Incidentally, the follies and vices of the worst side of society are exposed and castigated in the manner of Father Vaughan. Mrs. L. T. Meade has never written with greater effect than in this poignant story. A WOMAN'S AYE OR NAY By Lucas Cleeve All who are interested in the suffragette movement—and who is not?—will read Lucas Cleeve's new novel with profit and pleasure. The story is set some ten years ahead, when women are allowed to vote for Parliament; but although there is much in the novel of a quasi-political character, it is the love side of it which is uppermost, and which will call for highest appreciation. As an exponent of the tender passion, few living novelists can compare with Lucas Cleeve. VALDORA By Thomas Pinkerton Valdora belongs to the order of romance which is a perpetual joy to the novel reader. A Princess of a small State secures the services of an Englishman to defend her possessions from the attacks of envious neighbours. There is the clash of
- 43. arms, and the delight of love. Valdora suggests the method of Mr. Anthony Hope, with whose work it will well bear comparison. A WIFE FROM THE FORBIDDEN LAND By Archer Philip Crouch The particular fascination of Mr. A. P. Crouch's new story is that the scene of its operations is placed in that weird, mysterious land, Thibet. A young Englishman of the self-reliant, strong, and adventurous type determines to visit Lhasa—the sacred Thibetan capital—a city which the foreigner is not allowed to explore upon pain of death. How the Englishman succeeds in his object, and how he brings back with him A Wife from the Forbidden Land, is the function of the story to tell. Mr. Crouch knows the peoples of the wonderful East like a native; and his book is not only an engrossing romance: it is a vivid presentment of the customs, institutions, and manners of a land which is as yet but little known to the European. A YANKEE NAPOLEON By John F. Macpherson The Yankee Napoleon is a scientist who manufactures a brain serum, by which his own intellect and will-power are so enormously increased that the whole of America lies helpless at his feet. He uses his power, not like a benevolent genius, but like a criminal lunatic who is held in check by no law, human or divine. How, after a devastating war, in which East and West are involved, the Yankee Napoleon's plans are frustrated and brought to nothingness by an English scientist and a Japanese Marquis, the reader must discover for himself. He is confidently promised a story unique in plot and inventive power, full of amazing thrills, and written with the pen of a wizard. A FULL-LENGTH PORTRAIT OF EVE By Daniel Douglas Brewer The action of this story is placed in Paris, and the characters are wholly French, but A Full-Length Portrait of Eve will appeal with irresistible force to English readers. In its essence it is a love-story—a fiery, passionate, overwhelming love- story; and it is written with a beauty of phrase and a distinctive style rarely to be found in the work of a new writer. THE JEWEL HOUSE By Mrs. Isabel Smith Mrs. Isabel Smith has already achieved a reputation by her first work, The Minister's Guest. In her new story, The Jewel House, the same qualities of quiet charm and literary style will be found united to a fine gift of portraiture. The incidents happen in the country, and the chief characters are a baronet, the scion of an ancient house, and a young and beautiful girl of the yeoman class. The
- 44. Jewel House is a love-story; but it is natural, healthy, and wholly delightful, and it cannot fail to win fresh admirers for an authoress whose work has only to be known to be justly appreciated. IN SEARCH OF JÉHANNE By Avis Hekking A new historical romance which can bring effectively before the reader the life and manners of a bygone age, and make real flesh and blood of the characters which it introduces, is a rarity, and sure of a warm welcome. Such is In Search of Jéhanne. Miss Avis Hekking has taken the sixteenth century and the Massacre of the Huguenots as a background. The characters are French, and the events happen wholly in France. In Search of Jéhanne is a romance of first-rate quality, and it should create for the author a high position among writers of historical novels. A LIGHT-HEARTED REBELLION By John Langfield In this story the author unwinds a plot in which there is nothing hackneyed or commonplace, but which in character and incident is fresh and natural and wholesome, and brimming with delightful comedy. Humorists are rare, and readers will assuredly be glad to have their attention directed to this light, bright, laughable, captivating book. THE WAY OF WAR By Hew Scot Of late there have been attempts (feeble, it must be admitted) to portray in fiction the jealousy of Continental nations against our realm, but in no case has one of them approached the realistic and thrilling description of the sudden and secretly planned attempted invasion by Germany of Great Britain, as set forth in Hew Scot's splendid work, The Way of War. Commencing with the admittance of a German patient into an Edinburgh hospital, and the suggestion of a strong love interest, we are carried swiftly through seven days of intense stress and adventure, during which the fate of our country seems to hang on the skill and resource of a single individual, by whose adroitness and courage the secret designs of the enemy are discovered, and as far as possible prepared for, until the climax is reached in a naval battle, when the steel-clad might of England, being at length let loose, swoops down in all its tremendous power and sweeps the foe from our waters. This is a book full of life and movement, and one it is impossible to lay aside, having once commenced. ONE EVENTFUL SUMMER By Ethel Grace Tapner
- 45. One Eventful Summer is the work of a new writer, but not since the days of Lorna Doone has a story been written which conveys so much of the subtle charm of Devonshire. The central idea of the story is whimsical, but there is comedy and tragedy in it—love and laughter and tears. One Eventful Summer differs essentially from fiction in general, and upon that account, as well as upon its undoubted intrinsic merits, it will be greatly appreciated by those who are fortunate enough to read the book. A HUMAN BACILLUS By Robert Eustace A story that will make some demand upon the nerves of the reader, and leave behind it a burning remembrance. Such is A Human Bacillus. It describes the life and love of a strange being—partly genius, partly saint, and partly madman— whose subtle acts of renunciation and revenge lead to an extraordinary dénouement. The story is written by Robert Eustace, who is well known as the collaborator of L. T. Meade in The Brotherhood of the Seven Kings, The Sanctuary Club, The Sorceress of the Strand, etc. INCAPABLE LOVERS, LIMITED By W. W. Ward A new work of humour is rare in these days, and if the humour is of the right sort —if it is wholesome and natural and unforced—such a book is something to be thankful for. Incapable Lovers is written in great good spirits, and the characters and incidents sparkle and bubble over with delightful fun. Not since the days of Three Men in a Boat has so popular and infectious a work of humour appeared. THE SHADE OF THE ACACIA By Jessie E. Livesay To say that The Shade of the Acacia fulfils the high promise which was held out in the author's previous story, The Little Tin Gods, does not adequately describe the fine qualities of this novel. It not only shows literary gifts of an unusual order, but there are flashes of insight and penetration in it of rare power. The plot is taken up with the marriage of a baronet to a young girl who loves her husband's best friend; and the eternal duel between love and duty is presented in an entirely original form; even the minor characters are distinct and individual. In a word, The Shade of the Acacia belongs to the higher rank of fiction. THE SACRED HERB By Fergus Hume This story deals with a herb, brought from the South Seas, which is used for religious ceremonial, and is called the Devil Root by the natives. When burnt, the fumes produce insensibility, and release the spirit, which can see all that takes
- 46. place on the physical world, without being able to interfere. The hero is a witness (while in such a trance) to a murder, for which he is afterwards blamed; but he cannot prove his innocence, until assisted by the heroine, whose psychic powers enable her to clear his character. The book is a new departure for Mr. Fergus Hume, as, although the mystery is still retained, the tale deals largely with the power of the occult in modern life. THE AUTHOR WITH THE LARGEST PUBLIC The Sales of Nat Gould's Novels exceed 5,000,000 (five million) Copies NAT GOULD'S NEW NOVELS Mr. JOHN LONG is now the exclusive Publisher of all Mr. Nat Gould's New Novels, with cover designs in four colours by Mr. Harington Bird, the well-known horse painter. The following is the List to October, 1907: Price 2s. each, illustrated boards; or in cloth gilt, 2s. 6d. each. Crown 8vo., 288 pages ONE OF A MOB Ready THE SELLING PLATER Ready THE LADY TRAINER Ready A STRAIGHT GOER Ready A HUNDRED TO ONE CHANCE Ready A SPORTING SQUATTER Ready
- 47. CHARGER AND CHASER Ready THE CHANCE OF A LIFE-TIME Ready THE LITTLE WONDER July THE TOP WEIGHT October FIFTH YEAR OF PUBLICATION NAT GOULD'S ANNUAL, 1907 A STROKE OF LUCK Price 1s., large demy 8vo., 160 pages, sewed, cover in colours ⁂ Orders are now being taken for NAT GOULD'S ANNUAL for 1907 Ready in October NAT GOULD'S NEW SIXPENNY NOVELS In large demy 8vo., sewed. Striking cover in colours ONE OF A MOB Ready THE SELLING PLATER Ready A BIT OF A ROGUE Ready THE LADY TRAINER Ready A STRAIGHT GOER June A HUNDRED TO ONE CHANCE August A REPLY TO THE NEW THEOLOGY BY THE AUTHOR OF WHEN IT WAS LIGHT THE FOOL HATH SAID ——
- 48. The fool hath said in his heart, 'There is no God' Crown 8vo., paper cover, 1s. net; or in cloth gilt, 2s. 6d. net A real triumph of modern publishing.—Pall Mall Gazette A marvel of cheapness.—Spectator JOHN LONG'S LIBRARY OF MODERN CLASSICS A series of great works of fiction by modern authors. Not pocket editions, but large, handsome, and fully-illustrated volumes for the bookshelf, printed in large type on the best paper. Biographical Introductions and Photogravure Portraits. Size, 8 in. by 5-1/2 in.; thickness, 1-1/4 in. Prices: Cloth Gilt, 2s. net each; Leather, gold blocked and silk marker, 3s. net each; or in Classic Half-Vellum, 5s. net each. THE THREE CLERKS (480 pp.) Anthony Trollope THE CLOISTER AND THE HEARTH (672 pp.) Charles Reade THE WOMAN IN WHITE (576 pp.) Wilkie Collins ADAM BEDE (480 pp.) George Eliot THE HISTORY OF HENRY ESMOND (432 pp.) W. M. Thackeray WESTWARD HO! (600 pp.) Charles Kingsley TOM BROWN'S SCHOOLDAYS (320 pp.) Thomas Hughes A TALE OF TWO CITIES (384 pp.) Charles Dickens Other Volumes to follow.
- 49. SHILLING NET EDITIONS OF THREE FAMOUS BOOKS Each in Crown 8vo., thread sewn, printed on superior antique wove paper. With beautiful cover designs in colour by Charles E. Dawson CONFESSIONS OF A PRINCESS Being the astounding Revelations of Manners and Morals in European Courts. The late editor of the Saturday Review writes: The book gives one an extraordinary impression of reality; it is true, truer even than the shorthand account of a trial in our Divorce Court. I have enjoyed the book, and I will maintain before all and sundry that it is a good book, an excellent book, a book that had to be written. N.B.—The Sales of this remarkable book in the more Expensive Edition exceeded 200,000 copies. WILHELMINA IN LONDON By Barry Pain, Author of Eliza, etc. Standard. —A most amusing and delightful book. Take it along in the train, and you won't have to struggle for sleep against comfortless cushions and exiguous seats. Pall Mall Gazette.—A light, charming piece of literary frivolity. MR. AND MRS. VILLIERS By Hubert Wales, Author of The Yoke
- 50. The Times: The situations and the dialogue are handled with sureness and skill, and the two sisters present feminine character studies of singular beauty.—Daily Telegraph: Hubert Wales is a capable writer, and has produced a story which is worth being read.—Daily Chronicle: The story is extremely well written, the characterization admirable. Mr. Wales has amused us, and we have enjoyed his book.—Morning Leader: There is no denying the cleverness of the book.— Athenæum: Powerfully written.—Tatler: A very notable book. GENERAL LITERATURE A BOOK OF THE CEVENNES By S. Baring-Gould, M.A. With upwards of 40 Illustrations on art paper, printed in sepia, with 8 plates in colours and a Map. Crown 8vo., cloth gilt, price 6s. ⁂ This work is uniform in scope and size with the author's well- known books on Devon, Cornwall, Dartmoor, Brittany, the Riviera, etc. Prospectus post free PICTURES FROM NATURE'S GARDEN Or, Stories from Life in Wood and Field. By H. W. Shepheard-Walwyn, M.A., F.Z.S., F.E.S., etc. Author of Nature's Nursery, Nature's Riddles, The Lay of the Wee Brown Wren, etc. With 78 Illustrations on art paper, printed in sepia, from the author's photographs direct from Nature. Crown 8vo., cloth gilt, 6s. Prospectus post free TERRIERS: Their Points and Management By Frank Townend Barton, M.R.C.V.S.
- 51. With upwards of 40 Illustrations from photographs on art paper, printed in sepia. Crown 8vo., cloth, heavily gilt, price 5s. net. [Prospectus post free] This is an entirely new and important work on Terriers, and the only one of its kind yet published. It is a departure from the style usually followed by authors in canine literature. In a word, it is a book for everybody who wants to know all about terriers. THREE NEW VOLUMES OF POEMS In royal 16mo, cloth gilt and gilt top, with silk marker, price 3s. 6d. net each. THE TESTAMENT OF OMAR KHAYYAM (The Wasiyyat), comprising his Testament (or Last Words), A Song, Hymn of Prayer, The Word in the Desert, Hymn of Praise, also the Maráthi, or Odes of the Disciples. By Louis C. Alexander THE COMING OF SPRING; and Other Poems By Rowe Lingston VOCES AMORIS By John B. Rankin Will outbid all rivals.—The Bookman. Certainly wonderful.—Athenæum. It has remained for Mr. John Long to undersell all rivals by his 'Carlton Classics.' The copy before us—Thackeray's 'English
- 52. Humorists'—is extremely well printed and nicely got up, and must certainly be reckoned as the last word in cheap editions.—Daily News. JOHN LONG'S CARLTON CLASSICS Prices: Artistic Cloth, gilt, 6d. net; Leather, gilt top, gold-blocked back and side, 1s. net; postage, 1-1/2d. per vol. Length from 160 to 320 pages, newly set in clear, new type, and printed on the best paper. Each Volume contains a Biographical Introduction by the Editor, Mr. Hannaford Bennett. The first twelve only are bound in decorative paper covers. 1. THE FOUR GEORGES W. M. Thackeray 2. CHILDE HAROLD'S PILGRIMAGE Lord Byron 3. MUCH ADO ABOUT NOTHING Shakespeare 4. WARREN HASTINGS Lord Macaulay 5. THE LIFE OF NELSON Robert Southey 6. TALES (Selected) Edgar Allan Poe 7. CHRISTABEL, and other Poems S. T. Coleridge 8. A SENTIMENTAL JOURNEY Laurence Sterne 9. THE BLESSED DAMOZEL, and other Poems Dante Gabriel Rossetti 10. ON HEROES AND HERO WORSHIP Thomas Carlyle 11. SONNETS AND POEMS Shakespeare 12. RASSELAS Samuel Johnson 13. SONNETS AND POEMS Edmund Spenser 14. ESSAYS (Selected) Joseph Addison 15. HIS BOOK Artemus Ward 16. THE DUNCIAD, and other Poems Alexander Pope 17. ENGLISH HUMORISTS OF THE EIGHTEENTH CENTURY W. M. Thackeray
- 53. 18. THE JUMPING FROG, and Other Sketches Mark Twain 19. SONGS Robert Burns 20. ESSAYS (Selected) Leigh Hunt 21. LETTERS OF JUNIUS Anonymous 22. HUMOROUS POEMS Thomas Hood 23. CONFESSIONS OF AN ENGLISH OPIUM EATER Thomas De Quincey 24. A VOYAGE TO LILLIPUT Dean Swift 25. GRACE ABOUNDING John Bunyan 26. ESSAYS Matthew Arnold 27. POEMS Percy Bysshe Shelley 28. MR. GILFIN'S LOVE STORY George Eliot 29. SCENES FROM LORREQUER Charles Lever 30. POEMS Ben Jonson 31. COUNSELS CIVIL AND MORAL Francis Bacon 32. MINOR POEMS John Milton 33. SELECTIONS Edmund Burke 34. SONNETS William Wordsworth 35. A VOYAGE TO LISBON Henry Fielding 36. ESSAYS James Anthony Froude Other Volumes in Preparation
- 54. THE HAYMARKET NOVELS A Series of Copyright Novels by Popular Authors. The Volumes are printed upon a superior Antique Wove Paper, handsomely bound in specially designed cover, red cloth, heavily Gold Blocked at back. The size of the volumes is 7-1/2 in. by 5-1/4 in. by 1-1/4 in., the length from 300 to 350 pages, and the price 2s. 6d. each. VOLUMES NOW READY FATHER ANTHONY (Illustrated) Robert Buchanan A CABINET SECRET (Illustrated) Guy Boothby AN OUTSIDER'S YEAR Florence Warden FUGITIVE ANNE Mrs. Campbell Praed THE FUTURE OF PHYLLIS Adeline Sergeant BENEATH THE VEIL Adeline Sergeant THE SCARLET SEAL Dick Donovan AN ILL WIND Mrs. Lovett Cameron MIDSUMMER MADNESS Mrs. Lovett Cameron THE SILENT HOUSE IN PIMLICO Fergus Hume THE CRIMSON CRYPTOGRAM Fergus Hume A TRAITOR IN LONDON Fergus Hume THE MACHINATIONS OF JANET Sarah Tytler THE MAGNETIC GIRL Richard Marsh A FLIRTATION WITH TRUTH Curtis Yorke DELPHINE Curtis Yorke THE OTHER MRS. JACOBS Mrs. Campbell Praed A JILT'S JOURNAL Rita THE INDISCRETION OF GLADYS Lucas Cleeve PARTNERS THREE May Crommelin
- 55. ONCE TOO OFTEN Florence Warden THE LADY OF THE ISLAND (Illustrated) Guy Boothby THE WORLD MASTERS George Griffith HIS MASTER PURPOSE Harold Bindloss TREWINNOT OF GUY'S Mrs. Coulson Kernahan MRS. MUSGRAVE AND HER HUSBAND Richard Marsh THE CRIME OF THE CENTURY Dick Donovan CURIOS: or, the Strange Adventures of Two Bachelors (Illustrated) Richard Marsh A BRIDE FROM THE SEA Guy Boothby ROSAMOND GRANT Mrs. Lovett Cameron THE JADE EYE Fergus Hume MISS ARNOTT'S MARRIAGE Richard Marsh AN IMPOSSIBLE HUSBAND Florence Warden THE GIRL IN GREY CURTIS Yorke IN SPITE OF THE CZAR (Illustrated) Guy Boothby SAINT ELIZABETH OF LONDON Lucas Cleeve THE OPAL SERPENT Fergus Hume THE MAID OF THE RIVER Mrs. Campbell Praed THE FACE IN THE FLASHLIGHT Florence Warden ALIX OF THE GLEN Curtis Yorke VOLUMES IN PREPARATION THE SECRET PASSAGE Fergus Hume OLIVE KINSELLA Curtis Yorke ADVENTURES OF MIRANDA L. T. Meade
- 56. WATERS OF OBLIVION Adeline Sergeant THE MASK William Le Queux CONFESSIONS OF A YOUNG LADY Richard Marsh THE NIGHT OF RECKONING Frank Barrett RECENT POPULAR NOVELS SIX SHILLINGS EACH THE WORLD AND DELIA Curtis Yorke THE PENNILESS MILLIONAIRE David Christie Murray THE DUST OF CONFLICT Harold Bindloss THE HOUSE IN THE CRESCENT Adeline Sergeant THE DUKE'S DILEMMA Sir Wm. Magnay, Bart. THE MISTRESS OF AYDON R. H. Forster THE DUCHESS OF PONTIFEX SQUARE G. W. Appleton IZELLE OF THE DUNES C. Guise Mitford THE LUCK OF THE LEURA Mrs. Campbell Praed THE SWEETS OF OFFICE Violet Tweedale THE YOKE (Author of Mr. and Mrs. Villiers) Hubert Wales THE ROMANCE OF A MAID OF HONOUR Richard Marsh SELMA Lucas Cleeve
- 57. THE MAN WITH THE AMBER EYES Florence Warden THE HOUSE OF HOWE May Crommelin AMAZEMENT James Blyth THE MILLIONAIRE AND THE LADY Gertrude Warden KINDRED SPIRITS L. T. Meade A BUTTERFLY Baroness von Goldäcker DEYNCOURT OF DEYNCOURT Daisy Hugh Pryce THE FOLLY OF THE WISE G. Sidney Paternoster INNOCENT MASQUERADERS Sarah Tytler THE END OF A PASSION Alice M. Diehl THE SECOND EVIL Sadi Grant THE TWO FORCES E. Way Elkington ALL THAT A MAN HATH Coralie Stanton and Heath Hosken THE DICTIONARY OF FOOLS G. G. Chatterton TWO WOMEN AND A MAHARAJAH Mrs. C. E. Phillimore A SERPENT IN HIS WAY Suzanne Somers A MINISTER OF FATE Charles Dawson LEONE Lady Dunbar of Mochrum THE STAIN ON THE SHIELD Mrs. Darent Harrison A BEGGAR ON HORSEBACK S. R. Keightley FROM THE HAND OF THE HUNTER L. T. Meade THE GIRLS OF INVERBARNS Sarah Tytler JOHN LONG'S SIXPENNY NOVELS In Striking Picture Covers, 9 in. by 6 in.
- 58. NEW VOLUMES (JANUARY-SEPTEMBER, 1907) 58 A BRIDE FROM THE SEA Guy Boothby 59 WHEN IT WAS LIGHT (A Reply to When It was Dark) Well-known Author 60 A BIT OF A ROGUE Nat Gould 61 THE GIRL IN GREY Curtis Yorke 62 HIS ITALIAN WIFE Lucas Cleeve 63 THE LADY TRAINER Nat Gould 64 THE SECRET PASSAGE Fergus Hume 65 IN SPITE OF THE CZAR Guy Boothby 66 A STRAIGHT GOER Nat Gould 67 AN INNOCENT IMPOSTOR Maxwell Gray 68 THE STORM OF LONDON F. Dickberry 69 A LOST CAUSE Guy Thorne 70 ONE HUNDRED TO ONE CHANCE Nat Gould 71 FUGITIVE ANNE Mrs. Campbell Praed 72 THE NIGHT OF RECKONING Frank Barrett 1 AN OUTSIDER'S YEAR Florence Warden 2 SOMETHING IN THE CITY Florence Warden 3 THE LOVELY MRS. PEMBERTON Florence Warden 4 THE MYSTERY OF DUDLEY HORNE Florence Warden 5 KITTY'S ENGAGEMENT Florence Warden 6 OUR WIDOW Florence Warden 7 No. 3, THE SQUARE Florence Warden 8 THE JADE EYE Fergus Hume
- 59. 9 THE TURNPIKE HOUSE Fergus Hume 10 THE GOLDEN WANG-HO Fergus Hume 11 THE SILENT HOUSE IN PIMLICO Fergus Hume 12 THE CRIMSON CRYPTOGRAM Fergus Hume 13 A TRAITOR IN LONDON Fergus Hume 14 WOMAN—THE SPHINX Fergus Hume 15 A WOMAN'S NO Mrs. Lovett Cameron 16 THE CRAZE OF CHRISTINA Mrs. Lovett Cameron 17 A PASSING FANCY Mrs. Lovett Cameron 18 BITTER FRUIT Mrs. Lovett Cameron 19 AN ILL WIND Mrs. Lovett Cameron 20 MIDSUMMER MADNESS Mrs. Lovett Cameron 22 MRS. MUSGRAVE AND HER HUSBAND Richard Marsh 23 ADA VERNHAM, ACTRESS Richard Marsh 24 THE MAGNETIC GIRL Richard Marsh 25 MISS ARNOTT'S MARRIAGE Richard Marsh 26 THE MASK William Le Queux 27 THE EYE OF ISTAR William Le Queux 28 THE VEILED MAN William Le Queux 29 A MAN OF TO-DAY Helen Mathers 30 THE SIN OF HAGAR Helen Mathers 31 THE JUGGLER AND THE SOUL Helen Mathers 32 FATHER ANTHONY Robert Buchanan 33 THE WOOING OF MONICA L. T. Meade 34 THE BURDEN OF HER YOUTH L. T. Meade
- 60. 35 A JILT'S JOURNAL Rita 36 THE SIN OF JASPER STANDISH Rita 37 A CABINET SECRET Guy Boothby 38 THE LADY OF THE ISLAND Guy Boothby 39 THE FUTURE OF PHYLLIS Adeline Sergeant 40 BENEATH THE VEIL Adeline Sergeant 41 DELPHINE Curtis Yorke 42 THE COUNTESS OF MOUNTENOY John Strange Winter 43 THE SELLING PLATER Nat Gould 44 ONE OF A MOB Nat Gould 45 THE OTHER MRS. JACOBS Mrs. Campbell Praed 46 THE FLUTE OF PAN John Oliver Hobbes 47 THE KINGDOM OF MAMMON Violet Tweedale 48 THE STOLEN EMPEROR Mrs. Hugh Fraser 49 A BEAUTIFUL REBEL Ernest Glanville 51 THE WORLD MASTERS George Griffith 52 IN SUMMER SHADE Mary E. Mann 53 LE SELVE Ouida 54 SWEET DOLL OF HADDON HALL J. E. Muddock 55 GEORGE AND SON Edward H. Cooper 56 THE SCARLET SEAL Dick Donovan 57 THE THREE DAYS' TERROR J. S. Fletcher JOHN LONG, 12, 13 14 Norris Street, Haymarket, London
- 61. BILLING AND SONS, LIMITED, PRINTERS, GUILDFORD
- 62. Transcriber's Note: Minor punctuation errors have been corrected without note. Punctuation and spelling were made consistent when a predominant form was found in this book; otherwise they were not changed. Ambiguous hyphens at the ends of lines were retained. Footnotes were moved to the end of chapters and numbered in one continuous sequence. Mid-paragraph illustrations have been moved between paragraphs and some illustrations have been moved closer to the text that references them. The list of Illustrations paginations were changed accordingly. Other corrections: p. 14: Footnote 2: 'Nos' de Lavène replaced by 'Mos' de Lavène. p. 63: 'beseiged' replaced by 'besieged' (In 1590 he besieged Espaly again). p. 69: 'protusion' replaced by 'protrusion' (upon their protrusion from the volcanic vent). p. 76: 'irresitible' replaced by 'irrestistible' (proved an irresistible attraction). p. 79: 'pronouned replaced by 'pronounced.' (pronounced absolution). p. 94: 'aleep' replaced by 'asleep' (thinking that I was asleep). p. 100: 'Payrabeille' replaced by 'Peyrebeille' (on his way to Peyrebeille). p. 121: 'clientelle' replaced by 'clientele'(they have also their clientele).
- 63. p. 128: 'wonful' replaced by 'wonderful' (it is really wonderful what changes). p. 83 Footnote 5: 'Vivavais' replaced by 'Vivarais' (Pays Volcaniques du Vivarais). p. 139: 'abrubtly' replaced by 'abruptly' (the crag that falls abruptly). p. 156: 'Grott' replaced by 'Grotte' (reach the Grotte of the Chouans). p. 197: 'insurgeuts' replaced by 'insurgents' (when the insurgents saw). p. 225: 'Christian' replaced by 'Christina' (Queen Christina of Sweden). Pp. 248, 306: 'Valerauge' replaced by 'Vallerauge.' p. 264: 'Courbeson' replaced by 'Courbezon' (He produced Les Courbezon in 1862). p. 264: 'Orbe' replaced by 'Orb' (Bédarieux is on the Orb). p. 305: 'Belzunce' replaced by 'Belsunce' (Belsunce, Mgr. de, 174). p. 305: 'Chassizac' replaced by 'Chassezac' (Chassezac, River, 139, 154). p. 308: 'Rouve' replaced by 'Roure' (Roure, Baron de, 198). p. 308: 'Hippolite-du-Fort' replaced by 'Hippolyte-le-Fort' (S. Hippolite-le-Fort, 223- 4). Advertisements, p. 24: Number 21 in the list is missing. Also, the sequence starting with no. 1 on p. 24 is continued on p. 23 with no. 58. These were not corrected.
- 64. *** END OF THE PROJECT GUTENBERG EBOOK A BOOK OF THE CEVENNES *** Updated editions will replace the previous one—the old editions will be renamed. Creating the works from print editions not protected by U.S. copyright law means that no one owns a United States copyright in these works, so the Foundation (and you!) can copy and distribute it in the United States without permission and without paying copyright royalties. Special rules, set forth in the General Terms of Use part of this license, apply to copying and distributing Project Gutenberg™ electronic works to protect the PROJECT GUTENBERG™ concept and trademark. Project Gutenberg is a registered trademark, and may not be used if you charge for an eBook, except by following the terms of the trademark license, including paying royalties for use of the Project Gutenberg trademark. If you do not charge anything for copies of this eBook, complying with the trademark license is very easy. You may use this eBook for nearly any purpose such as creation of derivative works, reports, performances and research. Project Gutenberg eBooks may be modified and printed and given away—you may do practically ANYTHING in the United States with eBooks not protected by U.S. copyright law. Redistribution is subject to the trademark license, especially commercial redistribution. START: FULL LICENSE
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