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Nonholonomic Noetherian symmetries and integrals of the Routh sphere and
Chaplygin ball: The dynamics of a spherical body with a non-uniform mass distribution rolling
on a plane were discussed by Sergey Chaplygin, whose 150th anniversary we
celebrate this year. The Chaplygin top is a non-integrable system, with a
colourful range of interesting motions. A special case of this system was
studied by Edward Routh, who showed that it is integrable.
The Routh sphere has centre of mass offset from the geometric centre, but it
has an axis of symmetry through both these points, and equal moments of inertia
about all axes orthogonal to the symmetry axis. There are three constants of
motion: the total energy and two quantities involving the angular momenta.
It is straightforward to demonstrate that these quantities, known as the
Jellett and Routh constants, are integrals of the motion. However, their
physical significance has not been fully understood. In this paper, we show how
the integrals of the Routh sphere arise from Emmy Noether's invariance
identity. We derive expressions for the infinitesimal symmetry transformations
associated with these constants. We find the finite version of these symmetries
and provide their geometrical interpretation.
As a further demonstration of the power and utility of this method, we find
the Noether symmetries and corresponding Noether integrals for a system
introduced recently: the Chaplygin ball on a rotating turntable, confirming
that the known integrals are directly obtained from Noether's theorem. | nlin_CD |
Irreversibility of symbolic time series: a cautionary tale: Many empirical time series are genuinely symbolic: examples range from link
activation patterns in network science, DNA coding or firing patterns in
neuroscience to cryptography or combinatorics on words. In some other contexts,
the underlying time series is actually real-valued, and symbolization is
applied subsequently, as in symbolic dynamics of chaotic systems. Among several
time series quantifiers, time series irreversibility (the difference between
forward and backward statistics in stationary time series) is of great
relevance. However, the irreversible character of symbolized time series is not
always equivalent to the one of the underlying real-valued signal, leading to
some misconceptions and confusion on interpretability. Such confusion is even
bigger for binary time series (a classical way to encode chaotic trajectories
via symbolic dynamics). In this article we aim to clarify some usual
misconceptions and provide theoretical grounding for the practical analysis --
and interpretation -- of time irreversibility in symbolic time series. We
outline sources of irreversibility in stationary symbolic sequences coming from
frequency asymmetries of non-palindromic pairs which we enumerate, and prove
that binary time series cannot show any irreversibility based on words of
length m < 4, thus discussing the implications and sources of confusion. We
also study irreversibility in the context of symbolic dynamics, and clarify why
these can be reversible even when the underlying dynamical system is not, such
as the case of the fully chaotic logistic map. | nlin_CD |
Driven-Dissipative Dynamics of Atomic Ensembles in a Resonant Cavity II:
Quasiperiodic Route to Chaos and Chaotic Synchronization: We analyze the origin and properties of the chaotic dynamics of two atomic
ensembles in a driven-dissipative experimental setup, where they are
collectively damped by a bad cavity mode and incoherently pumped by a Raman
laser. Starting from the mean-field equations, we explain the emergence of
chaos by way of quasiperiodicity -- presence of two or more incommensurate
frequencies. This is known as the Ruelle-Takens-Newhouse route to chaos. The
equations of motion have a $\mathbb{Z}_{2}$-symmetry with respect to the
interchange of the two ensembles. However, some of the attractors of these
equations spontaneously break this symmetry. To understand the emergence and
subsequent properties of various attractors, we concurrently study the
mean-field trajectories, Poincar\'{e} sections, maximum and conditional
Lyapunov exponents, and power spectra. Using Floquet analysis, we show that
quasiperiodicity is born out of non $\mathbb{Z}_{2}$-symmetric oscillations via
a supercritical Neimark-Sacker bifurcation. Changing the detuning between the
level spacings in the two ensembles and the repump rate results in the
synchronization of the two chaotic ensembles. In this regime, the chaotic
intensity fluctuations of the light radiated by the two ensembles are
identical. Identifying the synchronization manifold, we understand the origin
of synchronized chaos as a tangent bifurcation intermittency of the
$\mathbb{Z}_{2}$-symmetric oscillations. At its birth, synchronized chaos is
unstable. The interaction of this attractor with other attractors causes on-off
intermittency until the synchronization manifold becomes sufficiently
attractive. We also show coexistence of different phases in small pockets near
the boundaries. | nlin_CD |
Large deviations in chaotic systems: exact results and dynamical phase
transition: Large deviations in chaotic dynamics have potentially significant and
dramatic consequences. We study large deviations of series of finite lengths
$N$ generated by chaotic maps. The distributions generally display an
exponential decay with $N$, associated with large-deviation (rate) functions.
We obtain the exact rate functions analytically for the doubling, tent, and
logistic maps. For the latter two, the solution is given as a power series
whose coefficients can be systematically calculated to any order. We also
obtain the rate function for the cat map numerically, uncovering strong
evidence for the existence of a remarkable singularity of it that we interpret
as a second order dynamical phase transition. Furthermore, we develop a
numerical tool for efficiently simulating atypical realizations of sequences if
the chaotic map is not invertible, and we apply it to the tent and logistic
maps. | nlin_CD |
Stochastic heating of a molecular nanomagnet: We study the excitation dynamics of a single molecular nanomagnet by static
and pulsed magnetic fields. Based on a stability analysis of the classical
magnetization dynamics we identify analytically the fields parameters for which
the energy is stochastically pumped into the system in which case the
magnetization undergoes diffusively and irreversibly a large angle deflection.
An approximate analytical expression for the diffusion constant in terms of the
fields parameters is given and assessed by full numerical calculations. | nlin_CD |
Information processing by a controlled coupling process: This Letter proposes a controlled coupling process for information
processing. The net effect of conventional coupling is isolated from the
dynamical system and is analyzed in depth. The stability of the process is
studied. We show that the proposed process can locally minimize the smoothness
and the fidelity of dynamical data. A digital filter expression of the
controlled coupling process is derived and the connection is made to the
Hanning filter. The utility and robustness of proposed approach is demonstrated
by both the restoration of the contaminated solution of the nonlinear
Schr\"{o}dinger equation and the estimation of the trend of a time series. | nlin_CD |
Linear differential equations to solve nonlinear mechanical problems: A
novel approach: Often a non-linear mechanical problem is formulated as a non-linear
differential equation. A new method is introduced to find out new solutions of
non-linear differential equations if one of the solutions of a given non-linear
differential equation is known. Using the known solution of the non-linear
differential equation, linear differential equations are set up. The solutions
of these linear differential equations are found using standard techniques.
Then the solutions of the linear differential equations are put into non-linear
differential equations and checked whether these solutions are also solutions
of the original non-linear differential equation. It is found that many
solutions of the linear differential equations are also solutions of the
original non-linear differential equation. | nlin_CD |
Weakly nonergodic dynamics in the Gross--Pitaevskii lattice: The microcanonical Gross--Pitaevskii (aka semiclassical Bose-Hubbard) lattice
model dynamics is characterized by a pair of energy and norm densities. The
grand canonical Gibbs distribution fails to describe a part of the density
space, due to the boundedness of its kinetic energy spectrum. We define
Poincare equilibrium manifolds and compute the statistics of microcanonical
excursion times off them. The tails of the distribution functions quantify the
proximity of the many-body dynamics to a weakly-nonergodic phase, which occurs
when the average excursion time is infinite. We find that a crossover to
weakly-nonergodic dynamics takes place inside the nonGibbs phase, being
unnoticed by the largest Lyapunov exponent. In the ergodic part of the
non-Gibbs phase, the Gibbs distribution should be replaced by an unknown
modified one. We relate our findings to the corresponding integrable limit,
close to which the actions are interacting through a short range coupling
network. | nlin_CD |
On the Amplitude of External Perurbation and Chaos via Devil's
Staircasein Muthuswamy-Chua System: We recently analyzed the voltage of the memristic circuit proposed by
Muthuswamy and Chua by adding an external sinusoidal oscillation $\gamma\omega
\cos\omega t$ to the ${\dot y}(t)\simeq {\dot i_L}(t)$, when the ${\dot
x}(t)\simeq {\dot v_C}(t)$ is given by $y(t)/C$.
When $f_s<f_d$ we have observed that the H\"older exponent of the system with
$C=1$ is larger than 1, and that of the system with $C=1.2$ is less than 1. The
latter system is unstable, and the route to chaos via the devil's staircase is
observed.
Above the mode of $f_d=1, f_s=1$ observed at $\omega\simeq 0.5$, we observed
a mode of $f_d=1, f_s=2$ at $\omega\simeq 1.15$ and $\simeq 1.05$, in the case
of $C=1$ and 1.2, respectively, and a mode of $f_d=2, f_s=3$ at $\omega\simeq
0.85$ and $\simeq 0.78$, in the case of $C=1$ and 1.2, respectively. At high
frequency of $f_s$, there is no qualitative difference in the stability of the
oscillation for $C=1$ and $C=1.2$ | nlin_CD |
Suppressing chaos in discontinuous systems of fractional order by active
control: In this paper, a chaos control algorithm for a class of piece-wise continuous
chaotic systems of fractional order, in the Caputo sense, is proposed. With the
aid of Filippov's convex regularization and via differential inclusions, the
underlying discontinuous initial value problem is first recast in terms of a
set-valued problem and hence it is continuously approximated by using Cellina's
Theorem for differential inclusions. For chaos control, an active control
technique is implemented so that the unstable equilibria become stable. As
example, Shimizu--Morioka's system is considered. Numerical simulations are
obtained by means of the Adams-Bashforth-Moulton method for differential
equations of fractional-order. | nlin_CD |
Periodic orbit quantization of a Hamiltonian map on the sphere: In a previous paper we introduced examples of Hamiltonian mappings with phase
space structures resembling circle packings. It was shown that a vast number of
periodic orbits can be found using special properties. We now use this
information to explore the semiclassical quantization of one of these maps. | nlin_CD |
Quadratically damped oscillators with non-linear restoring force: In this paper we qualitatively analyse quadratically damped oscillators with
non-linear restoring force. In particular, we obtain Hamiltonian structure and
analytical form of the energy functions. | nlin_CD |
Search for conformal invariance in compressible two-dimensional
turbulence: We present a search for conformal invariance in vorticity isolines of
two-dimensional compressible turbulence. The vorticity is measured by tracking
the motion of particles that float at the surface of a turbulent tank of water.
The three-dimensional turbulence in the tank has a Taylor microscale
$Re_\lambda \simeq 160$. The conformal invariance theory being tested here is
related to the behavior of equilibrium systems near a critical point. This
theory is associated with the work of L\"owner, Schramm and others and is
usually referred to as Schramm-L\"owner Evolution (SLE). The system was exposed
to several tests of SLE. The results of these tests suggest that zero-vorticity
isolines exhibit noticeable departures from this type of conformal invariance. | nlin_CD |
Exponential Fermi acceleration in general time-dependent billiards: It is shown, that under very general conditions, a generic time-dependent
billiard, for which a phase-space of corresponding static (frozen) billiards is
of the mixed type, exhibits the exponential Fermi acceleration in the adiabatic
limit. The velocity dynamics in the adiabatic regime is represented as an
integral over a path through the abstract space of invariant components of
corresponding static billiards, where the paths are generated probabilistically
in terms of transition-probability matrices. We study the statistical
properties of possible paths and deduce the conditions for the exponential
Fermi acceleration. The exponential Fermi acceleration and theoretical concepts
presented in the paper are demonstrated numerically in four different
time-dependent billiards. | nlin_CD |
Connecting period-doubling cascades to chaos: The appearance of infinitely-many period-doubling cascades is one of the most
prominent features observed in the study of maps depending on a parameter. They
are associated with chaotic behavior, since bifurcation diagrams of a map with
a parameter often reveal a complicated intermingling of period-doubling
cascades and chaos. Period doubling can be studied at three levels of
complexity. The first is an individual period-doubling bifurcation. The second
is an infinite collection of period doublings that are connected together by
periodic orbits in a pattern called a cascade. It was first described by
Myrberg and later in more detail by Feigenbaum. The third involves infinitely
many cascades and a parameter value $\mu_2$ of the map at which there is chaos.
We show that often virtually all (i.e., all but finitely many) ``regular''
periodic orbits at $\mu_2$ are each connected to exactly one cascade by a path
of regular periodic orbits; and virtually all cascades are either paired --
connected to exactly one other cascade, or solitary -- connected to exactly one
regular periodic orbit at $\mu_2$. The solitary cascades are robust to large
perturbations. Hence the investigation of infinitely many cascades is
essentially reduced to studying the regular periodic orbits of $F(\mu_2,
\cdot)$. Examples discussed include the forced-damped pendulum and the
double-well Duffing equation. | nlin_CD |
Statistical properties of $r$-adic processes and their connections to
families of popular fractal curves: Results concerning the statists of $r$-adic processes and their fractal
properties are reviewed. The connection between singular eigenstates of the
statistical evolution of such processes and popular fractal curves is
emphasized. | nlin_CD |
Characteristics of in-out intermittency in delay-coupled FitzHugh-Nagumo
oscillators: We analyze a pair of delay-coupled FitzHugh-Nagumo oscillators exhibiting
in-out intermittency as a part of the generating mechanism of extreme events.
We study in detail the characteristics of in-out intermittency and identify the
invariant subsets involved --- a saddle fixed point and a saddle periodic orbit
--- neither of which are chaotic as in the previously reported cases of in-out
intermittency. Based on the analysis of a periodic attractor possessing in-out
dynamics, we can characterize the approach to the invariant synchronization
manifold and the spiralling out to the saddle periodic orbit with subsequent
ejection from the manifold. Due to the striking similarities, this analysis of
in-out dynamics explains also in-out intermittency. | nlin_CD |
Power fluctuations in a driven damped chaotic pendulum: In this paper we investigate the power fluctuations in a driven, dampted
pendulum. When the motion of the pendulum is chaotic, the average power
supplied by the driving force is equal to the average dissipated power only for
an infinite long time period. We measure the fluctuations of the supplied power
during a time equal to the period of the driving force. Negative power
fluctuations occur and we estimate their probability. In a chaotic state the
histogram of the power distribution is broad and continuous although bounded.
For a value of the power not too close to the edge of the distribution the
Fluctuation Theorem of Gallavotti and Cohen is approximately satisfied. | nlin_CD |
Birth and Death of Chimera: Interplay of Delay and Multiplexing: The chimera state with co-existing coherent-incoherent dynamics has recently
attracted a lot of attention due to its wide applicability. We investigate
non-locally coupled identical chaotic maps with delayed interactions in the
multiplex network framework and find that an interplay of delay and
multiplexing brings about an enhanced or suppressed appearance of chimera state
depending on the distribution as well as the parity of delay values in the
layers. Additionally, we report a layer chimera state with an existence of one
layer displaying coherent and another layer demonstrating incoherent dynamical
evolution. The rich variety of dynamical behavior demonstrated here can be used
to gain further insight into the real-world networks which inherently possess
such multi-layer architecture with delayed interactions. | nlin_CD |
Uniform semiclassical wave function for coherent 2D electron flow: We find a uniform semiclassical (SC) wave function describing coherent
branched flow through a two-dimensional electron gas (2DEG), a phenomenon
recently discovered by direct imaging of the current using scanned probed
microscopy. The formation of branches has been explained by classical
arguments, but the SC simulations necessary to account for the coherence are
made difficult by the proliferation of catastrophes in the phase space. In this
paper, expansion in terms of "replacement manifolds" is used to find a uniform
SC wave function for a cusp singularity. The method is then generalized and
applied to calculate uniform wave functions for a quantum-map model of coherent
flow through a 2DEG. Finally, the quantum-map approximation is dropped and the
method is shown to work for a continuous-time model as well. | nlin_CD |
Linear response, susceptibility and resonances in chaotic toy models: We consider simple examples illustrating some new features of the linear
response theory developed by Ruelle for dissipative and chaotic systems [{\em
J. of Stat. Phys.} {\bf 95} (1999) 393]. In this theory the concepts of linear
response, susceptibility and resonance, which are familiar to physicists, have
been revisited due to the dynamical contraction of the whole phase space onto
attractors. In particular the standard framework of the
"fluctuation-dissipation" theorem breaks down and new resonances can show up
oustside the powerspectrum. In previous papers we proposed and used new
numerical methods to demonstrate the presence of the new resonances predicted
by Ruelle in a model of chaotic neural network. In this article we deal with
simpler models which can be worked out analytically in order to gain more
insights into the genesis of the ``stable'' resonances and their consequences
on the linear response of the system. We consider a class of 2-dimensional
time-discrete maps describing simple rotator models with a contracting radial
dynamics onto the unit circle and a chaotic angular dynamics $\theta_{t+1} = 2
\theta_t (\mod 2\pi)$. A generalisation of this system to a network of
interconnected rotators is also analysed and related with our previous studies
\cite{CS1,CS2}. These models permit us to classify the different types of
resonances in the susceptibility and to discuss in particular the relation
between the relaxation time of the system to equilibrium with the {\em mixing}
time given by the decay of the correlation functions. Also it enables one to
propose some general mechanisms responsible for the creation of stable
resonances with arbitrary frequencies, widths, and dependency on the pair of
perturbed/observed variables. | nlin_CD |
Periodic orbit analysis at the onset of the unstable dimension
variability and at the blowout bifurcation: Many chaotic dynamical systems of physical interest present a strong form of
nonhyperbolicity called unstable dimension variability (UDV), for which the
chaotic invariant set contains periodic orbits possessing different numbers of
unstable eigendirections. The onset of UDV is usually related to the loss of
transversal stability of an unstable fixed point embedded in the chaotic set.
In this paper, we present a new mechanism for the onset of UDV, whereby the
period of the unstable orbits losing transversal stability tends to infinity as
we approach the onset of UDV. This mechanism is unveiled by means of a periodic
orbit analysis of the invariant chaotic attractor for two model dynamical
systems with phase spaces of low dimensionality, and seems to depend heavily on
the chaotic dynamics in the invariant set. We also described, for these
systems, the blowout bifurcation (for which the chaotic set as a whole loses
transversal stability) and its relation with the situation where the effects of
UDV are the most intense. For the latter point, we found that chaotic
trajectories off, but very close to, the invariant set exhibit the same scaling
characteristic of the so-called on-off intermittency. | nlin_CD |
On the Asymptotics of the Hopf Characteristic Function: We study the asymptotic behavior of the Hopf characteristic function of
fractals and chaotic dynamical systems in the limit of large argument. The
small argument behavior is determined by the moments, since the characteristic
function is defined as their generating function. Less well known is that the
large argument behavior is related to the fractal dimension. While this
relation has been discussed in the literature, there has been very little in
the way of explicit calculation. We attempt to fill this gap, with explicit
calculations for the generalized Cantor set and the Lorenz attractor. In the
case of the generalized Cantor set, we define a parameter characterizing the
asymptotics which we show corresponds exactly to the known fractal dimension.
The Hopf characteristic function of the Lorenz attractor is computed
numerically, obtaining results which are consistent with Hausdorff or
correlation dimension, albeit too crude to distinguish between them. | nlin_CD |
Transient Chaos Generates Small Chimeras: While the chimera states themselves are usually believed to be chaotic
transients, the involvement of chaos behind their self-organization is not
properly distinguished or studied. In this work, we demonstrate that small
chimeras in the local flux dynamics of an array of magnetically coupled
superconducting quantum interference devices (SQUIDs) driven by an external
field are born through transiently chaotic dynamics. We deduce analytic
expressions for small chimeras and synchronous states which correspond to
nonchaotic attractors in the model. We also numerically study the bifurcations
underlying the multistability responsible for their generation. Transient chaos
manifests itself in the short term flux oscillations with erratically
fluctuating amplitudes, exponential escape time distribution and irregular
dependence of the escape time to initial conditions. We classify the small
chimera states in terms of the position of the non-synchronized member and
numerically construct their basin of attraction. The basin is shown to possess
an interesting structure consisting of both ordered and fractal parts, which
again can be attributed to transient chaos. | nlin_CD |
Analytical perturbative approach to periodic orbits in the homogeneous
quartic oscillator potential: We present an analytical calculation of periodic orbits in the homogeneous
quartic oscillator potential. Exploiting the properties of the periodic
Lam{\'e} functions that describe the orbits bifurcated from the fundamental
linear orbit in the vicinity of the bifurcation points, we use perturbation
theory to obtain their evolution away from the bifurcation points. As an
application, we derive an analytical semiclassical trace formula for the
density of states in the separable case, using a uniform approximation for the
pitchfork bifurcations occurring there, which allows for full semiclassical
quantization. For the non-integrable situations, we show that the uniform
contribution of the bifurcating period-one orbits to the coarse-grained density
of states competes with that of the shortest isolated orbits, but decreases
with increasing chaoticity parameter $\alpha$. | nlin_CD |
Visualizing Attractors of the Three-Dimensional Generalized Hénon
Map: We study dynamics of a generic quadratic diffeomorphism, a 3D generalization
of the planar H\'{e}non map. Focusing on the dissipative, orientation
preserving case, we give a comprehensive parameter study of codimension-one and
two bifurcations. Periodic orbits, born at resonant, Neimark-Sacker
bifurcations, give rise to Arnold tongues in parameter space. Aperiodic
attractors include invariant circles and chaotic orbits; these are
distinguished by rotation number and Lyapunov exponents. Chaotic orbits include
H\'{e}non-like and Lorenz-like attractors, which can arise from period-doubling
cascades, and those born from the destruction of invariant circles. The latter
lie on paraboloids near the local unstable manifold of a fixed point. | nlin_CD |
Driven response of time delay coupled limit cycle oscillators: We study the periodic forced response of a system of two limit cycle
oscillators that interact with each other via a time delayed coupling. Detailed
bifurcation diagrams in the parameter space of the forcing amplitude and
forcing frequency are obtained for various interesting limits using numerical
and analytical means. In particular, the effects of the coupling strength, the
natural frequency spread of the two oscillators and the time delay parameter on
the size and nature of the entrainment domain are delineated. The system is
found to display a nonlinear response on certain critical contours in the space
of the coupling strength and time delay. Time delay offers a novel tuning knob
for controlling the system response over a wide range of frequencies and this
may have important practical applications. | nlin_CD |
Exploring Isomerization Dynamics on a Potential Energy Surface with an
Index-2 Saddle using Lagrangian Descriptors: In this paper we explore the phase space structures governing isomerization
dynamics on a potential energy surface with four wells and an index-2 saddle.
For this model, we analyze the influence that coupling both degrees of freedom
of the system and breaking the symmetry of the problem have on the geometrical
template of phase space structures that characterizes reaction. To achieve this
goal we apply the method of Lagrangian descriptors, a technique with the
capability of unveiling the key invariant manifolds that determine transport
processes in nonlinear dynamical systems. This approach reveals with
extraordinary detail the intricate geometry of the isomerization routes
interconnecting the different potential wells, and provides us with valuable
information to distinguish between initial conditions that undergo sequential
and concerted isomerization. | nlin_CD |
Integrable Approximation of Regular Islands: The Iterative Canonical
Transformation Method: Generic Hamiltonian systems have a mixed phase space, where classically
disjoint regions of regular and chaotic motion coexist. We present an iterative
method to construct an integrable approximation, which resembles the regular
dynamics of a given mixed system and extends it into the chaotic region. The
method is based on the construction of an integrable approximation in action
representation which is then improved in phase space by iterative applications
of canonical transformations. This method works for strongly perturbed systems
and arbitrary degrees of freedom. We apply it to the standard map and the
cosine billiard. | nlin_CD |
Probability Distribution of the Quality Factor of a Mode-Stirred
Reverberation Chamber: We derive a probability distribution, confidence intervals and statistics of
the quality (Q) factor of an arbitrarily shaped mode-stirred reverberation
chamber, based on ensemble distributions of the idealized random cavity field
with assumed perfect stir efficiency. It is shown that Q exhibits a
Fisher-Snedecor F-distribution whose degrees of freedom are governed by the
number of simultaneously excited cavity modes per stir state. The most probable
value of Q is between a fraction 2/9 and 1 of its mean value, and between a
fraction 4/9 and 1 of its asymptotic (composite Q) value. The arithmetic mean
value is found to always exceed the values of all other theoretical metrics for
centrality of Q. For a rectangular cavity, we retrieve the known asymptotic Q
in the limit of highly overmoded regime. | nlin_CD |
Finite-time synchronization between two different chaotic systems with
uncertainties: A new method of virtual unknown parameter is proposed to synchronize two
different systems with unknown parameters and disturbance in finite time.
Virtual unknown parameters are introduced in order to avoid the unknown
parameters from appearing in the controllers and parameters update laws when
the adaptive control method is applied. A single virtual unknown parameter is
used in the design of adaptive controllers and parameters update laws if the
Lipschitz constant on the nonlinear function can be found, while multiple
virtual unknown parameters are adopted if the Lipschitz constant cannot be
determined. Numerical simulations show that the present method does make the
two different chaotic systems synchronize in finite time. | nlin_CD |
Uniform approximation of barrier penetration in phase space: A method to approximate transmission probabilities for a nonseparable
multidimensional barrier is applied to a waveguide model. The method uses
complex barrier-crossing orbits to represent reaction probabilities in phase
space and is uniform in the sense that it applies at and above a threshold
energy at which classical reaction switches on. Above this threshold the
geometry of the classically reacting region of phase space is clearly reflected
in the quantum representation. Two versions of the approximation are applied. A
harmonic version which uses dynamics linearised around an instanton orbit is
valid only near threshold but is easy to use. A more accurate and more widely
applicable version using nonlinear dynamics is also described. | nlin_CD |
Are generalized synchronization and noise--induced synchronization
identical types of synchronous behavior of chaotic oscillators?: This paper deals with two types of synchronous behavior of chaotic
oscillators -- generalized synchronization and noise--induced synchronization.
It has been shown that both these types of synchronization are caused by
similar mechanisms and should be considered as the same type of the chaotic
oscillator behavior. The mechanisms resulting in the generalized
synchronization are mostly similar to ones taking place in the case of the
noise-induced synchronization with biased noise. | nlin_CD |
Statistical Theory of Magnetohydrodynamic Turbulence: Recent Results: In this review article we will describe recent developments in statistical
theory of magnetohydrodynamic (MHD) turbulence. Kraichnan and Iroshnikov first
proposed a phenomenology of MHD turbulence where Alfven time-scale dominates
the dynamics, and the energy spectrum E(k) is proportional to k^{-3/2}. In the
last decade, many numerical simulations show that spectral index is closer to
5/3, which is Kolmogorov's index for fluid turbulence. We review recent
theoretical results based on anisotropy and Renormalization Groups which
support Kolmogorov's scaling for MHD turbulence.
Energy transfer among Fourier modes, energy flux, and shell-to-shell energy
transfers are important quantities in MHD turbulence. We report recent
numerical and field-theoretic results in this area. Role of these quantities in
magnetic field amplification (dynamo) are also discussed. There are new
insights into the role of magnetic helicity in turbulence evolution. Recent
interesting results in intermittency, large-eddy simulations, and shell models
of magnetohydrodynamics are also covered. | nlin_CD |
The analysis of restricted five-body problem within frame of variable
mass: In the framework of restricted five bodies problem, the existence and
stability of the libration points are explored and analysed numerically, under
the effect of non--isotropic mass variation of the fifth body (test particle or
infinitesimal body). The evolution of the positions of these points and the
possible regions of motion are illustrated, as a function of the perturbation
parameter. We perform a systematic investigation in an attempt to understand
how the perturbation parameter due to variable mass of the fifth body, affects
the positions, movement and stability of the libration points. In addition, we
have revealed how the domain of the basins of convergence associated with the
libration points are substantially influenced by the perturbation parameter. | nlin_CD |
Chaotic advection and targeted mixing: The advection of passive tracers in an oscillating vortex chain is
investigated. It is shown that by adding a suitable perturbation to the ideal
flow, the induced chaotic advection exhibits two remarkable properties compared
with a generic perturbation : Particles remain trapped within a specific domain
bounded by two oscillating barriers (suppression of chaotic transport along the
channel), and the stochastic sea seems to cover the whole domain (enhancement
of mixing within the rolls). | nlin_CD |
Chaos Pass Filter: Linear Response of Synchronized Chaotic Systems: The linear response of synchronized time-delayed chaotic systems to small
external perturbations, i.e., the phenomenon of chaos pass filter, is
investigated for iterated maps. The distribution of distances, i.e., the
deviations between two synchronized chaotic units due to external perturbations
on the transfered signal, is used as a measure of the linear response. It is
calculated numerically and, for some special cases, analytically. Depending on
the model parameters this distribution has power law tails in the region of
synchronization leading to diverging moments of distances. This is a
consequence of multiplicative and additive noise in the corresponding linear
equations due to chaos and external perturbations. The linear response can also
be quantified by the bit error rate of a transmitted binary message which
perturbs the synchronized system. The bit error rate is given by an integral
over the distribution of distances and is calculated analytically and
numerically. It displays a complex nonmonotonic behavior in the region of
synchronization. For special cases the distribution of distances has a fractal
structure leading to a devil's staircase for the bit error rate as a function
of coupling strength. The response to small harmonic perturbations shows
resonances related to coupling and feedback delay times. A bi-directionally
coupled chain of three units can completely filtered out the perturbation. Thus
the second moment and the bit error rate become zero. | nlin_CD |
A Study on the Synchronization Aspect of Star Connected Identical Chua
Circuits: This paper provides a study on the synchronization aspect of star connected
$N$ identical chua's circuits. Different coupling such as conjugate coupling,
diffusive coupling and mean-field coupling have been investigated in star
topology. Mathematical interpretation of different coupling aspects have been
explained. Simulation results of different coupling mechanism have been
studied. | nlin_CD |
Adaptive control of the singularly perturbed chaotic systems based on
the scale time estimation by keeping chaotic property: In this paper, a new approach to the problem of stabilizing a chaotic system
is presented. In this regard, stabilization is done by sustaining chaotic
properties of the system. Sustaining the chaotic properties has been mentioned
to be of importance in some areas such as biological systems. The problem of
stabilizing a chaotic singularly perturbed system will be addressed and a
solution will be proposed based on the OGY (Ott, Grebogi and Yorke)
methodology. For the OGY control, Poincare section of the system is defined on
its slow manifold. The multi-time scale property of the singularly perturbed
system is exploited to control the Poincare map with the slow scale time. Slow
scale time is adaptively estimated using a parameter estimation technique.
Control with slow time scale circumvents the need to observe the states. With
this strategy, the system remains chaotic and chaos identification is possible
with online calculation of lyapunov exponents. Using this strategy on
ecological system improves their control in three aspects. First that for
ecological systems sustaining the dynamical property is important to survival
of them. Second it removes the necessity of insertion of control action in each
sample time. And third it introduces the sufficient time for census. | nlin_CD |
Incommensurate standard map: We introduce and study the extension of the Chirikov standard map when the
kick potential has two and three incommensurate spatial harmonics. This system
is called the incommensurate standard map. At small kick amplitudes the
dynamics is bounded by the isolating Kolmogorov-Arnold-Moser surfaces while
above a certain kick strength it becomes unbounded and diffusive. The quantum
evolution at small quantum kick amplitudes is somewhat similar to the case of
Aubru-Andr\'e model studied in mathematics and experiments with cold atoms in a
static incommensurate potential. We show that for the quantum map there is also
a metal-insulator transition in space while in momentum we have localization
similar to the case of 2D Anderson localization. In the case of three
incommensurate frequencies of space potential the quantum evolution is
characterized by the Anderson transition similar to 3D case of disordered
potential. We discuss possible physical systems with such map description
including dynamics of comets and dark matter in planetary systems. | nlin_CD |
Synchronization of Fractional-order Chaotic Systems with Gaussian
fluctuation by Sliding Mode Control: This paper is devoted to the problem of synchronization between
fractional-order chaotic systems with Gaussian fluctuation by the method of
fractional-order sliding mode control. A fractional integral (FI) sliding
surface is proposed for synchronizing the uncertain fractional-order system,
and then the sliding mode control technique is carried out to realize the
synchronization of the given systems. One theorem about sliding mode controller
is presented to prove the proposed controller can make the system synchronize.
As a case study, the presented method is applied to the fractional-order
Chen-L\"u system as the drive-response dynamical system. Simulation results
show a good performance of the proposed control approach in synchronizing the
chaotic systems in presence of Gaussian noise. | nlin_CD |
Basin entropy as an indicator of a bifurcation in a time-delayed system: The basin entropy is a measure that quantifies, in a system that has two or
more attractors, the predictability of a final state, as a function of the
initial conditions. While the basin entropy has been demonstrated on a variety
of multistable dynamical systems, to the best of our knowledge, it has not yet
been tested in systems with a time delay, whose phase space is infinite
dimensional because the initial conditions are functions defined in a time
interval $[-\tau,0]$, where $\tau$ is the delay time. Here we consider a simple
time delayed system consisting of a bistable system with a linear delayed
feedback term. We show that the basin entropy captures relevant properties of
the basins of attraction of the two coexisting attractors. Moreover, we show
that the basin entropy can give an indication of the proximity of a Hopf
bifurcation, but fails to capture the proximity of a pitchfork bifurcation. Our
results suggest that the basin entropy can yield useful insights into the
long-term predictability of time delayed systems, which often have coexisting
attractors. | nlin_CD |
Boundary Circles of Mixed Phase Space, Hamiltonian Systems: The phase space of an area-preserving map typically contains infinitely many
elliptic islands embedded in a chaotic sea. Orbits near the boundary of a
chaotic region have been observed to stick for long times, strongly influencing
their transport properties. The boundary is composed of invariant circles,
called "Boundary circles." We investigate the distribution of rotation numbers
of boundary circles for the Henon quadratic map and show that the probability
of occurrence of small elements of their continued fraction expansions is
larger than would be expected for a number chosen at random. However, large
elements occur with probabilities distributed proportionally to the random
case. These results have implications for models of transport in mixed phase
space. | nlin_CD |
Harvesting entropy and quantifying the transition from noise to chaos in
a photon-counting feedback loop: Some physical processes, including the intensity fluctuations of a chaotic
laser, the detection of single photons, and the Brownian motion of a
microscopic particle in a fluid are unpredictable, at least on long timescales.
This unpredictability can be due to a variety of physical mechanisms, but it is
quantified by an entropy rate. This rate describes how quickly a system
produces new and random information, is fundamentally important in statistical
mechanics and practically important for random number generation. We
experimentally study entropy generation and the emergence of deterministic
chaotic dynamics from discrete noise in a system that applies feedback to a
weak optical signal at the single-photon level. We show that the dynamics
qualitatively change from shot noise to chaos as the photon rate increases, and
that the entropy rate can reflect either the deterministic or noisy aspects of
the system depending on the sampling rate and resolution. | nlin_CD |
Crisis and unstable dimension variability in the bailout embedding map: The dynamics of inertial particles in $2-d$ incompressible flows can be
modeled by $4-d$ bailout embedding maps. The density of the inertial particles,
relative to the density of the fluid, is a crucial parameter which controls the
dynamical behaviour of the particles. We study here the dynamical behaviour of
aerosols, i.e. particles heavier than the flow. An attractor widening and
merging crisis is seen the phase space in the aerosol case. Crisis induced
intermittency is seen in the time series and the laminar length distribution of
times before bursts gives rise to a power law with the exponent $\beta=-1/3$.
The maximum Lyapunov exponent near the crisis fluctuates around zero indicating
unstable dimension variability (UDV) in the system. The presence of unstable
dimension variability is confirmed by the behaviour of the probability
distributions of the finite time Lyapunov exponents. | nlin_CD |
Extreme events in solutions of hydrostatic and non-hydrostatic climate
models: Initially this paper reviews the mathematical issues surrounding the
hydrostatic (HPE) and non-hydrostatic (NPE) primitive equations that have been
used extensively in numerical weather prediction and climate modelling. Cao and
Titi (2005, 2007) have provided a new impetus to this by proving existence and
uniqueness of solutions of viscous HPE on a cylinder with Neumann-like boundary
conditions on the top and bottom. In contrast, the regularity of solutions of
NPE remains an open question. With this HPE regularity result in mind, the
second issue examined in this paper is whether extreme events are allowed to
arise spontaneously in their solutions. Such events could include, for example,
the sudden appearance and disappearance of locally intense fronts that do not
involve deep convection. Analytical methods are used to show that for viscous
HPE, the creation of small-scale structures is allowed locally in space and
time at sizes that scale inversely with the Reynolds number. | nlin_CD |
Stretching and folding diagnostics in solutions of the three-dimensional
Euler and Navier-Stokes equations: Two possible diagnostics of stretching and folding (S&F) in fluid flows are
discussed, based on the dynamics of the gradient of potential vorticity ($q =
\bom\cdot\nabla\theta$) associated with solutions of the three-dimensional
Euler and Navier-Stokes equations. The vector $\bdB = \nabla q \times
\nabla\theta$ satisfies the same type of stretching and folding equation as
that for the vorticity field $\bom $ in the incompressible Euler equations
(Gibbon & Holm, 2010). The quantity $\theta$ may be chosen as the potential
temperature for the stratified, rotating Euler/Navier-Stokes equations, or it
may play the role of a seeded passive scalar for the Euler equations alone. The
first discussion of these S&F-flow diagnostics concerns a numerical test for
Euler codes and also includes a connection with the two-dimensional surface
quasi-geostrophic equations. The second S&F-flow diagnostic concerns the
evolution of the Lamb vector $\bsD = \bom\times\bu$, which is the nonlinearity
for Euler's equations apart from the pressure. The curl of the Lamb vector
($\boldsymbol{\varpi} := \bsD$) turns out to possess similar stretching and
folding properties to that of the $\bdB$-vector. | nlin_CD |
Estimation of System Parameters and Predicting the Flow Function from
Time Series of Continuous Dynamical Systems: We introduce a simple method to estimate the system parameters in continuous
dynamical systems from the time series. In this method, we construct a modified
system by introducing some constants (controlling constants) into the given
(original) system. Then the system parameters and the controlling constants are
determined by solving a set of nonlinear simultaneous algebraic equations
obtained from the relation connecting original and modified systems. Finally,
the method is extended to find the form of the evolution equation of the system
itself. The major advantage of the method is that it needs only a minimal
number of time series data and is applicable to dynamical systems of any
dimension. The method also works extremely well even in the presence of noise
in the time series. This method is illustrated for the case of Lorenz system. | nlin_CD |
Time-delayed model of immune response in plants: In the studies of plant infections, the plant immune response is known to
play an essential role. In this paper we derive and analyse a new mathematical
model of plant immune response with particular account for post-transcriptional
gene silencing (PTGS). Besides biologically accurate representation of the PTGS
dynamics, the model explicitly includes two time delays to represent the
maturation time of the growing plant tissue and the non-instantaneous nature of
the PTGS. Through analytical and numerical analysis of stability of the steady
states of the model we identify parameter regions associated with recovery and
resistant phenotypes, as well as possible chronic infections. Dynamics of the
system in these regimes is illustrated by numerical simulations of the model. | nlin_CD |
Renormalization group in the statistical theory of turbulence: Two-loop
approximation: The field theoretic renormalization group is applied to the stochastic
Navier--Stokes equation that describes fully developed fluid turbulence. The
complete two-loop calculation of the renormalization constant, the beta
function and the fixed point is performed. The ultraviolet correction exponent,
the Kolmogorov constant and the inertial-range skewness factor are derived to
second order of the $\epsilon$ expansion. | nlin_CD |
Fluctuation of similarity (FLUS) to detect transitions between distinct
dynamical regimes in short time series: Recently a method which employs computing of fluctuations in a measure of
nonlinear similarity based on local recurrence properties in a univariate time
series, was introduced to identify distinct dynamical regimes and transitions
between them in a short time series [1]. Here we present the details of the
analytical relationships between the newly introduced measure and the well
known concepts of attractor dimensions and Lyapunov exponents. We show that the
new measure has linear dependence on the effective dimension of the attractor
and it measures the variations in the sum of the Lyapunov spectrum. To
illustrate the practical usefulness of the method, we employ it to identify
various types of dynamical transitions in different nonlinear models. Also, we
present testbed examples for the new method's robustness against the presence
of noise and missing values in the time series. Furthermore, we use this method
to analyze time series from the field of social dynamics, where we present an
analysis of the US crime record's time series from the year 1975 to 1993. Using
this method, we have found that dynamical complexity in robberies was
influenced by the unemployment rate till late 1980's. We have also observed a
dynamical transition in homicide and robbery rates in the late 1980's and early
1990's, leading to increase in the dynamical complexity of these rates. | nlin_CD |
Limiter Control of a Chaotic RF Transistor Oscillator: We report experimental control of chaos in an electronic circuit at 43.9 MHz,
which is the fastest chaos control reported in the literature to date. Limiter
control is used to stabilize a periodic orbit in a tuned collector transistor
oscillator modified to exhibit simply folded band chaos. The limiter is
implemented using a transistor to enable monitoring the relative magnitude of
the control perturbation. A plot of the relative control magnitude vs. limiter
level shows a local minimum at period-1 control, thereby providing strong
evidence that the controlled state is an unstable periodic orbit (UPO) of the
uncontrolled system. | nlin_CD |
Nodal domains on quantum graphs: We consider the real eigenfunctions of the Schr\"odinger operator on graphs,
and count their nodal domains. The number of nodal domains fluctuates within an
interval whose size equals the number of bonds $B$. For well connected graphs,
with incommensurate bond lengths, the distribution of the number of nodal
domains in the interval mentioned above approaches a Gaussian distribution in
the limit when the number of vertices is large. The approach to this limit is
not simple, and we discuss it in detail. At the same time we define a random
wave model for graphs, and compare the predictions of this model with analytic
and numerical computations. | nlin_CD |
Oscillation quenching in third order Phase Locked Loop coupled by mean
field diffusive coupling: We explore analytically the oscillation quenching phenomena (amplitude death
and oscillation death) in a coupled third order phase locked loop (PLL) both in
periodic and chaotic mode. The phase locked loops are coupled through mean
field diffusive coupling. The lower and upper limits of the quenched state are
identified in the parameter space of the coupled PLL using Routh-Hurwitz
technique. We further observe that the ability of convergence to the quenched
state of coupled PLLs depends on the design parameters. For identical system
both the system converges to homogeneous steady state whereas for non-identical
parameter values they converge to inhomogeneous steady state. It is also
observed that for identical systems the quenched state is wider than
non-identical case. When the systems parameters are so chosen that each
isolated loops are chaotic in nature, in that case we observe the quenched
state is relatively narrow. All these phenomena are also demonstrated through
numerical simulations. | nlin_CD |
Photoabsorption spectra of the diamagnetic hydrogen atom in the
transition regime to chaos: Closed orbit theory with bifurcating orbits: With increasing energy the diamagnetic hydrogen atom undergoes a transition
from regular to chaotic classical dynamics, and the closed orbits pass through
various cascades of bifurcations. Closed orbit theory allows for the
semiclassical calculation of photoabsorption spectra of the diamagnetic
hydrogen atom. However, at the bifurcations the closed orbit contributions
diverge. The singularities can be removed with the help of uniform
semiclassical approximations which are constructed over a wide energy range for
different types of codimension one and two catastrophes. Using the uniform
approximations and applying the high-resolution harmonic inversion method we
calculate fully resolved semiclassical photoabsorption spectra, i.e.,
individual eigenenergies and transition matrix elements at laboratory magnetic
field strengths, and compare them with the results of exact quantum
calculations. | nlin_CD |
Universal Power-law Decay in Hamiltonian Systems?: The understanding of the asymptotic decay of correlations and of the
distribution of Poincar\'e recurrence times $P(t)$ has been a major challenge
in the field of Hamiltonian chaos for more than two decades. In a recent
Letter, Chirikov and Shepelyansky claimed the universal decay $P(t) \sim
t^{-3}$ for Hamiltonian systems. Their reasoning is based on renormalization
arguments and numerical findings for the sticking of chaotic trajectories near
a critical golden torus in the standard map. We performed extensive numerics
and find clear deviations from the predicted asymptotic exponent of the decay
of $P(t)$. We thereby demonstrate that even in the supposedly simple case, when
a critical golden torus is present, the fundamental question of asymptotic
statistics in Hamiltonian systems remains unsolved. | nlin_CD |
Planar Visibility Graph Network Algorithm For Two Dimensional Timeseries: In this brief paper, a simple and fast computational method, the Planar
Visibility Graph Networks Algorithm was proposed based on the famous Visibility
Graph Algorithm, which can fulfill converting two dimensional timeseries into a
planar graph. The constructed planar graph inherits several properties of the
series in its structure. Thereby, periodic series, random series, and chaotic
series convert into quite different networks with different average degree,
characteristic path length, diameter, clustering coefficient, different degree
distribution, and modularity, etc. By means of this new approach, with such
different networks measures, one can characterize two dimensional timeseries
from a new viewpoint of complex networks. | nlin_CD |
The impact of hydrodynamic interactions on the preferential
concentration of inertial particles in turbulence: We consider a dilute gas of inertial particles transported by the turbulent
flow. Due to inertia the particles concentrate preferentially outside vortices.
The pair-correlation function of the particles' concentration is known to obey
at small separations a power-law with a negative exponent, if the hydrodynamic
interactions between the particles are neglected. The divergence at zero
separation is the signature of the random attractor asymptoted by the
particles' trajectories at large times. However the hydrodynamic interactions
produce a repulsion between the particles that is non-negligible at small
separations. We introduce equations governing the repulsion and show it
smoothens the singular attractor near the particles where the pair correlation
function saturates. The effect is most essential at the Stokes number of order
one, where the correlations decrease by a factor of a few. | nlin_CD |
An ergodic averaging method to differentiate covariant Lyapunov vectors: Covariant Lyapunov vectors or CLVs span the expanding and contracting
directions of perturbations along trajectories in a chaotic dynamical system.
Due to efficient algorithms to compute them that only utilize trajectory
information, they have been widely applied across scientific disciplines,
principally for sensitivity analysis and predictions under uncertainty. In this
paper, we develop a numerical method to compute the directional derivatives of
CLVs along their own directions. Similar to the computation of CLVs, the
present method for their derivatives is iterative and analogously uses the
second-order derivative of the chaotic map along trajectories, in addition to
the Jacobian. We validate the new method on a super-contracting Smale-Williams
Solenoid attractor. We also demonstrate the algorithm on several other examples
including smoothly perturbed Arnold Cat maps, and the Lorenz attractor,
obtaining visualizations of the curvature of each attractor. Furthermore, we
reveal a fundamental connection of the CLV self-derivatives with a statistical
linear response formula. | nlin_CD |
Classical dynamics on graphs: We consider the classical evolution of a particle on a graph by using a
time-continuous Frobenius-Perron operator which generalizes previous
propositions. In this way, the relaxation rates as well as the chaotic
properties can be defined for the time-continuous classical dynamics on graphs.
These properties are given as the zeros of some periodic-orbit zeta functions.
We consider in detail the case of infinite periodic graphs where the particle
undergoes a diffusion process. The infinite spatial extension is taken into
account by Fourier transforms which decompose the observables and probability
densities into sectors corresponding to different values of the wave number.
The hydrodynamic modes of diffusion are studied by an eigenvalue problem of a
Frobenius-Perron operator corresponding to a given sector. The diffusion
coefficient is obtained from the hydrodynamic modes of diffusion and has the
Green-Kubo form. Moreover, we study finite but large open graphs which converge
to the infinite periodic graph when their size goes to infinity. The lifetime
of the particle on the open graph is shown to correspond to the lifetime of a
system which undergoes a diffusion process before it escapes. | nlin_CD |
Meanders and Reconnection-Collision Sequences in the Standard Nontwist
Map: New global periodic orbit collision/separatrix reconnection scenarios in the
standard nontwist map in different regions of parameter space are described in
detail, including exact methods for determining reconnection thresholds that
are implemented numerically. The results are compared to a break-up diagram of
shearless invariant curves. The existence of meanders (invariant tori that are
not graphs) is demonstrated numerically for both odd and even period
reconnection for certain regions in parameter space, and some of the
implications on transport are discussed. | nlin_CD |
Parametric Generator of Robust Chaos: Circuit Implementation and
Simulation Using the Program Product MULTISIM: A scheme is suggested of the parametric generator of chaotic oscillations
with attractor represented by a kind of Smale-Williams solenoid that operates
under a periodic sequence of pump pulses at two different frequencies.
Simulation of chaotic dynamics using the software product Multisim is provided. | nlin_CD |
On the relation between reliable computation time, float-point precision
and the Lyapunov exponent in chaotic systems: The relation among reliable computation time, Tc, float-point precision, K,
and the Lyapunov exponent, {\lambda}, is obtained as Tc= (lnB/{\lambda})K+C,
where B is the base of the float-point system and C is a constant dependent
only on the chaotic equation. The equation shows good agreement with numerical
experimental results, especially the scale factors. | nlin_CD |
Phase Synchronization on Spacially Embeded Duplex Networks with Total
Cost Constraint: Synchronization on multiplex networks have attracted increasing attention in
the past few years. We investigate collective behaviors of Kuramoto oscillators
on single layer and duplex spacial networks with total cost restriction, which
was introduced by Li et. al [Li G., Reis S. D., Moreira A. A., Havlin S.,
Stanley H. E. and Jr A. J., {\it Phys. Rev. Lett.} 104, 018701 (2010)] and
termed as the Li network afterwards. In the Li network model, with the increase
of its spacial exponent, the network's structure will vary from the random type
to the small-world one, and finally to the regular lattice.We first explore how
the spacial exponent influences the synchronizability of Kuramoto oscillators
on single layer Li networks and find that the closer the Li network is to a
regular lattice, the more difficult for it to evolve into synchronization. Then
we investigate synchronizability of duplex Li networks and find that the
existence of inter-layer interaction can greatly enhance inter-layer and global
synchronizability. When the inter-layer coupling strength is larger than a
certain critical value, whatever the intra-layer coupling strength is, the
inter-layer synchronization will always occur. Furthermore, on single layer Li
networks, nodes with larger degrees more easily reach global synchronization,
while on duplex Li networks, this phenomenon becomes much less obvious.
Finally, we study the impact of inter-link density on global synchronization
and obtain that sparse inter-links can lead to the emergence of global
synchronization for duplex Li networks just as dense inter-links do. In a word,
inter-layer interaction plays a vital role in determining synchronizability for
duplex spacial networks with total cost constraint. | nlin_CD |
Constructing a chaotic system with any number of equilibria: In the chaotic Lorenz system, Chen system and R\"ossler system, their
equilibria are unstable and the number of the equilibria are no more than
three. This paper shows how to construct some simple chaotic systems that can
have any preassigned number of equilibria. First, a chaotic system with no
equilibrium is presented and discussed. Then, a methodology is presented by
adding symmetry to a new chaotic system with only one stable equilibrium, to
show that chaotic systems with any preassigned number of equilibria can be
generated. By adjusting the only parameter in these systems, one can further
control the stability of their equilibria. This result reveals an intrinsic
relationship of the global dynamical behaviors with the number and stability of
the equilibria of a chaotic system. | nlin_CD |
Spectral statistics in chaotic systems with a point interaction: We consider quantum systems with a chaotic classical limit that are perturbed
by a point-like scatterer. The spectral form factor K(tau) for these systems is
evaluated semiclassically in terms of periodic and diffractive orbits. It is
shown for order tau^2 and tau^3 that off-diagonal contributions to the form
factor which involve diffractive orbits cancel exactly the diagonal
contributions from diffractive orbits, implying that the perturbation by the
scatterer does not change the spectral statistic. We further show that
parametric spectral statistics for these systems are universal for small
changes of the strength of the scatterer. | nlin_CD |
Matrix logistic map: fractal spectral distributions and transfer of
chaos: The standard logistic map, $x'=ax(1-x)$, serves as a paradigmatic model to
demonstrate how apparently simple non-linear equations lead to complex and
chaotic dynamics. In this work we introduce and investigate its matrix analogue
defined for an arbitrary matrix $X$ of a given order $N$. We show that for an
arbitrary initial ensemble of hermitian random matrices with a continuous level
density supported on the interval $[0,1]$, the asymptotic level density
converges to the invariant measure of the logistic map. Depending on the
parameter $a$ the constructed measure may be either singular, fractal or
described by a continuous density. In a broader class of the map multiplication
by a scalar logistic parameter $a$ is replaced by transforming
$aX(\mathbb{I}-X)$ into $BX(\mathbb{I}-X)B^{\dagger}$, where $A=BB^{\dagger}$
is a fixed positive matrix of order $N$. This approach generalizes the known
model of coupled logistic maps, and allows us to study the transition to chaos
in complex networks and multidimensional systems. In particular, associating
the matrix $B$ with a given graph we demonstrate the gradual transfer of chaos
between subsystems corresponding to vertices of a graph and coupled according
to its edges. | nlin_CD |
Nilpotent normal form for divergence-free vector fields and
volume-preserving maps: We study the normal forms for incompressible flows and maps in the
neighborhood of an equilibrium or fixed point with a triple eigenvalue. We
prove that when a divergence free vector field in $\mathbb{R}^3$ has nilpotent
linearization with maximal Jordan block then, to arbitrary degree, coordinates
can be chosen so that the nonlinear terms occur as a single function of two
variables in the third component. The analogue for volume-preserving
diffeomorphisms gives an optimal normal form in which the truncation of the
normal form at any degree gives an exactly volume-preserving map whose inverse
is also polynomial inverse with the same degree. | nlin_CD |
Extensive packet excitations in FPU and Toda lattices: At low energies, the excitation of low frequency packets of normal modes in
the Fermi-Pasta-Ulam (FPU) and in the Toda model leads to exponentially
localized energy profiles which resemble staircases and are identified by a
slope $\sigma$ that depends logarithmically on the specific energy
$\varepsilon=E/N$. Such solutions are found to lie on stable lower dimensional
tori, named $q$-tori. At higher energies there is a sharp transition of the
system's localization profile to a straight-line one, determined by an
$N$-dependent slope of the form $\sigma \sim (\varepsilon N)^{-d}$, $d>0$. We
find that the energy crossover $\varepsilon_c$ between the two energy regimes
decays as $1/N$, which indicates that $q$-tori disappear in the thermodynamic
limit. Furthermore, we focus on the times that such localization profiles are
practically frozen and we find that these "stickiness times" can rapidly and
accurately distinguish between a power-law and a stretched exponential
dependence in $1/\varepsilon $. | nlin_CD |
Entry and exit sets in the dynamics of area preserving Henon map: In this paper we study dynamical properties of the area preserving Henon map,
as a discrete version of open Hamiltonian systems, that can exhibit chaotic
scattering. Exploiting its geometric properties we locate the exit and entry
sets, i.e. regions through which any forward, respectively backward, unbounded
orbit escapes to infinity. In order to get the boundaries of these sets we
prove that the right branch of the unstable manifold of the hyperbolic fixed
point is the graph of a function, which is the uniform limit of a sequence of
functions whose graphs are arcs of the symmetry lines of the Henon map, as a
reversible map. | nlin_CD |
Quasiperiodicity and suppression of multistability in nonlinear
dynamical systems: It has been known that noise can suppress multistability by dynamically
connecting coexisting attractors in the system which are otherwise in separate
basins of attraction. The purpose of this mini-review is to argue that
quasiperiodic driving can play a similar role in suppressing multistability. A
concrete physical example is provided where quasiperiodic driving was
demonstrated to eliminate multistability completely to generate robust chaos in
a semiconductor superlattice system. | nlin_CD |
Slow energy relaxation and localization in 1D lattices: We investigate the energy relaxation process produced by thermal baths at
zero temperature acting on the boundary atoms of chains of classical anharmonic
oscillators. Time-dependent perturbation theory allows us to obtain an explicit
solution of the harmonic problem: even in such a simple system nontrivial
features emerge from the interplay of the different decay rates of Fourier
modes. In particular, a crossover from an exponential to an inverse-square-root
law occurs on a time scale proportional to the system size $N$. A further
crossover back to an exponential law is observed only at much longer times (of
the order $N^3$). In the nonlinear chain, the relaxation process is initially
equivalent to the harmonic case over a wide time span, as illustrated by
simulations of the $\beta$ Fermi-Pasta-Ulam model. The distinctive feature is
that the second crossover is not observed due to the spontaneous appearance of
breathers, i.e. space-localized time-periodic solutions, that keep a finite
residual energy in the lattice. We discuss the mechanism yielding such
solutions and also explain why it crucially depends on the boundary conditions. | nlin_CD |
Chaos in Black holes Surrounded by Electromagnetic Fields: In this paper we prove the occurence of chaos for charged particles moving
around a Schwarzshild black hole, perturbed by uniform electric and magnetic
fields. The appearance of chaos is studied resorting to the Poincare'-Melnikov
method. | nlin_CD |
Chaos control in the fractional order logistic map via impulses: In this paper the chaos control in the discrete logistic map of fractional
order is obtained with an impulsive control algorithm. The underlying discrete
initial value problem of fractional order is considered in terms of Caputo
delta fractional difference. Every $\Delta$ steps, the state variable is
instantly modified with the same impulse value, chosen from a bifurcation
diagram versus impulse. It is shown that the solution of the impulsive control
is bounded. The numerical results are verified via time series, histograms, and
the 0-1 test. Several examples are considered | nlin_CD |
Selective amplification of scars in a chaotic optical fiber: In this letter we propose an original mechanism to select scar modes through
coherent gain amplification in a multimode D-shaped fiber. More precisely, we
numerically demonstrate how scar modes can be amplified by positioning a gain
region in the vicinity of specific points of a short periodic orbit known to
give rise to scar modes. | nlin_CD |
Outliers, Extreme Events and Multiscaling: Extreme events have an important role which is sometime catastrophic in a
variety of natural phenomena including climate, earthquakes and turbulence, as
well as in man-made environments like financial markets. Statistical analysis
and predictions in such systems are complicated by the fact that on the one
hand extreme events may appear as "outliers" whose statistical properties do
not seem to conform with the bulk of the data, and on the other hands they
dominate the (fat) tails of probability distributions and the scaling of high
moments, leading to "abnormal" or "multi"-scaling. We employ a shell model of
turbulence to show that it is very useful to examine in detail the dynamics of
onset and demise of extreme events. Doing so may reveal dynamical scaling
properties of the extreme events that are characteristic to them, and not
shared by the bulk of the fluctuations. As the extreme events dominate the
tails of the distribution functions, knowledge of their dynamical scaling
properties can be turned into a prediction of the functional form of the tails.
We show that from the analysis of relatively short time horizons (in which the
extreme events appear as outliers) we can predict the tails of the probability
distribution functions, in agreement with data collected in very much longer
time horizons. The conclusion is that events that may appear unpredictable on
relatively short time horizons are actually a consistent part of a multiscaling
statistics on longer time horizons. | nlin_CD |
The discontinuous dynamics and non-autonomous chaos: A multidimensional chaos is generated by a special initial value problem for
the non-autonomous impulsive differential equation. The existence of a chaotic
attractor is shown, where density of periodic solutions, sensitivity of
solutions and existence of a trajectory dense in the set of all orbits are
observed. The chaotic properties of all solutions are discussed. An appropriate
example is constructed, where the intermittency phenomenon is indicated. The
results of the paper are illustrating that impulsive differential equations may
play a special role in the investigation of the complex behavior of dynamical
systems, different from that played by continuous dynamics. | nlin_CD |
Scaling laws of passive tracer dispersion in the turbulent surface layer: Experimental results for passive tracer dispersion in the turbulent surface
layer under stable conditions are presented. In this case, the dispersion of
tracer particles is determined by the interplay of three mechanisms: relative
dispersion (celebrated Richardson's mechanism), shear dispersion (particle
separation due to variation of the mean velocity field) and specific
surface-layer dispersion (induced by the gradient of the energy dissipation
rate in the turbulent surface layer). The latter mechanism results in the
rather slow (ballistic) law for the mean squared particle separation. Based on
a simplified Langevin equation for particle separation we found that the
ballistic regime always dominates at large times. This conclusion is supported
by our extensive atmospheric observations. Exit-time statistics are derived
from the experimental dataset and show a reasonable match with the simple
dimensional asymptotes for different mechanisms of tracer dispersion, as well
as predictions of the multifractal model and experimental data from other
sources. | nlin_CD |
Delay-induced homoclinic bifurcations in modified gradient bistable
systems and their relevance to optimisation: Nonlinear dynamical systems with time delay are abundant in applications, but
are notoriously difficult to analyse and predict because delay-induced effects
strongly depend on the form of the nonlinearities involved, and on the exact
way the delay enters the system. We consider a special class of nonlinear
systems with delay obtained by taking a gradient dynamical system with a
two-well "potential" function and replacing the argument of the right-hand side
function with its delayed version. This choice of the system is motivated by
the relative ease of its graphical interpretation, and by its relevance to a
recent approach to use delay in finding the global minimum of a multi-well
function. Here, the simplest type of such systems is explored, for which we
hypothesise and verify the possibility to qualitatively predict the
delay-induced effects, such as a chain of homoclinic bifurcations one by one
eliminating local attractors and enabling the phase trajectory to spontaneously
visit vicinities of all local minima. The key phenomenon here is delay-induced
reorganisation of manifolds, which cease to serve as barriers between the local
minima after homoclinic bifurcations. Despite the general scenario being quite
universal in two-well potentials, the homoclinic bifurcation comes in various
versions depending on the fine features of the potential. Our results are a
pre-requisite for understanding general highly nonlinear multistable systems
with delay. They also reveal the mechanisms behind the possible role of delay
in optimisation. | nlin_CD |
Cluster synchronization in complex network of coupled chaotic circuits:
an experimental study: By a small-size complex network of coupled chaotic Hindmarsh-Rose circuits,
we study experimentally the stability of network synchronization to the removal
of shortcut links. It is shown that the removal of a single shortcut link may
destroy either completely or partially the network synchronization.
Interestingly, when the network is partially desynchronized, it is found that
the oscillators can be organized into different groups, with oscillators within
each group being highly synchronized but are not for oscillators from different
groups, showing the intriguing phenomenon of cluster synchronization. The
experimental results are analyzed by the method of eigenvalue analysis, which
implies that the formation of cluster synchronization is crucially dependent on
the network symmetries. Our study demonstrates the observability of cluster
synchronization in realistic systems, and indicates the feasibility of
controlling network synchronization by adjusting network topology. | nlin_CD |
Emerging attractors and the transition from dissipative to conservative
dynamics: The topological structure of basin boundaries plays a fundamental role in the
sensitivity to the initial conditions in chaotic dynamical systems. Herewith we
present a study on the dynamics of dissipative systems close to the Hamiltonian
limit, emphasising the increasing number of periodic attractors and on the
structural changes in their basin boundaries as the dissipation approaches
zero. We show numerically that a power law with nontrivial exponent describes
the growth of the total number of periodic attractors as the damping is
decreased. We also establish that for small scales the dynamics is governed by
\emph{effective} dynamical invariants, whose measure depends not only on the
region of the phase space, but also on the scale under consideration.
Therefore, our results show that the concept of effective invariants is also
relevant for dissipative systems. | nlin_CD |
Time dependence of moments of an exactly solvable Verhulst model under
random perturbations: Explicit expressions for one point moments corresponding to stochastic
Verhulst model driven by Markovian coloured dichotomous noise are presented. It
is shown that the moments are the given functions of a decreasing exponent. The
asymptotic behavior (for large time) of the moments is described by a single
decreasing exponent. | nlin_CD |
The conservative cascade of kinetic energy in compressible turbulence: The physical nature of compressible turbulence is of fundamental importance
in a variety of astrophysical settings. We present the first direct evidence
that mean kinetic energy cascades conservatively beyond a transitional
"conversion" scale-range despite not being an invariant of the compressible
flow dynamics. We use high-resolution three-dimensional simulations of
compressible hydrodynamic turbulence on $512^3$ and $1024^3$ grids. We probe
regimes of forced steady-state isothermal flows and of unforced decaying ideal
gas flows. The key quantity we measure is pressure dilatation cospectrum,
$E^{PD}(k)$, where we provide the first numerical evidence that it decays at a
rate faster than $k^{-1}$ as a function of wavenumber. This is sufficient to
imply that mean pressure dilatation acts primarily at large-scales and that
kinetic and internal energy budgets statistically decouple beyond a
transitional scale-range. Our results suggest that an extension of Kolmogorov's
inertial-range theory to compressible turbulence is possible. | nlin_CD |
Relation of stability and bifurcation properties between continuous and
ultradiscrete dynamical systems via discretization with positivity: one
dimensional cases: Stability and bifurcation properties of one-dimensional discrete dynamical
systems with positivity, which are derived from continuous ones by tropical
discretization, are studied. The discretized time interval is introduced as a
bifurcation parameter in the discrete dynamical systems, and emergence
condition of an additional bifurcation, flip bifurcation, is identified.
Correspondence between the discrete dynamical systems with positivity and the
ultradiscrete ones derived from them is discussed. It is found that the derived
ultradiscrete max-plus dynamical systems can retain the bifurcations of the
original continuous ones via tropical discretization and ultradiscretization. | nlin_CD |
Stochastic dynamics and control of a driven nonlinear spin chain: the
role of Arnold diffusion: We study a chain of non-linear, interacting spins driven by a static and a
time-dependent magnetic field. The aim is to identify the conditions for the
locally and temporally controlled spin switching. Analytical and full numerical
calculations show the possibility of stochastic control if the underlying
semi-classical dynamics is chaotic. This is achievable by tuning the external
field parameters according to the method described in this paper. We show
analytically for a finite spin chain that Arnold diffusion is the underlying
mechanism for the present stochastic control. Quantum mechanically we consider
the regime where the classical dynamics is regular or chaotic. For the latter
we utilize the random matrix theory. The efficiency and the stability of the
non-equilibrium quantum spin-states are quantified by the time-dependence of
the Bargmann angle related to the geometric phases of the states. | nlin_CD |
Semiclassical Quantization by Harmonic Inversion: Comparison of
Algorithms: Harmonic inversion techniques have been shown to be a powerful tool for the
semiclassical quantization and analysis of quantum spectra of both classically
integrable and chaotic dynamical systems. Various computational procedures have
been proposed for this purpose. Our aim is to find out which method is
numerically most efficient. To this end, we summarize and discuss the different
techniques and compare their accuracies by way of two example systems. | nlin_CD |
Large-scale lognormality in turbulence modeled by Ornstein-Uhlenbeck
process: Lognormality was found experimentally for coarse-grained squared turbulence
velocity and velocity increment when the coarsening scale is comparable to the
correlation scale of the velocity (Mouri et al. Phys. Fluids 21, 065107, 2009).
We investigate this large-scale lognormality by using a simple stochastic
process with correlation, the Ornstein-Uhlenbeck (OU) process. It is shown that
the OU process has a similar large-scale lognormality, which is studied
numerically and analytically. | nlin_CD |
Controlling spatiotemporal chaos in oscillatory reaction-diffusion
systems by time-delay autosynchronization: Diffusion-induced turbulence in spatially extended oscillatory media near a
supercritical Hopf bifurcation can be controlled by applying global time-delay
autosynchronization. We consider the complex Ginzburg-Landau equation in the
Benjamin-Feir unstable regime and analytically investigate the stability of
uniform oscillations depending on the feedback parameters. We show that a
noninvasive stabilization of uniform oscillations is not possible in this type
of systems. The synchronization diagram in the plane spanned by the feedback
parameters is derived. Numerical simulations confirm the analytical results and
give additional information on the spatiotemporal dynamics of the system close
to complete synchronization. | nlin_CD |
Chaos in three coupled rotators: From Anosov dynamics to hyperbolic
attractors: Starting from Anosov chaotic dynamics of geodesic flow on a surface of
negative curvature, we develop and consider a number of self-oscillatory
systems including those with hinged mechanical coupling of three rotators and a
system of rotators interacting through a potential function. These results are
used to design an electronic circuit for generation of rough (structurally
stable) chaos. Results of numerical integration of the model equations of
different degree of accuracy are presented and discussed. Also, circuit
simulation of the electronic generator is provided using the NI Multisim
environment. Portraits of attractors, waveforms of generated oscillations,
Lyapunov exponents, and spectra are considered and found to be in good
correspondence for the dynamics on the attractive sets of the self-oscillatory
systems and for the original Anosov geodesic flow. The hyperbolic nature of the
dynamics is tested numerically using a criterion based on statistics of angles
of intersection of stable and unstable subspaces of the perturbation vectors at
a reference phase trajectory on the attractor. | nlin_CD |
A model for shock wave chaos: We propose the following model equation:
\[u_{t}+1/2(u^{2}-uu_{s})_{x}=f(x,u_{s}), \] that predicts chaotic shock waves.
It is given on the half-line $x<0$ and the shock is located at $x=0$ for any
$t\ge0$. Here $u_{s}(t)$ is the shock state and the source term $f$ is assumed
to satisfy certain integrability constraints as explained in the main text. We
demonstrate that this simple equation reproduces many of the properties of
detonations in gaseous mixtures, which one finds by solving the reactive Euler
equations: existence of steady traveling-wave solutions and their instability,
a cascade of period-doubling bifurcations, onset of chaos, and shock formation
in the reaction zone. | nlin_CD |
Recursive Tangential-Angular Operator as Analyzer of Synchronized Chaos: A method for the quantitative analysis of the degree and parameters of
synchronization of the chaotic oscillations in two coupled oscillators is
proposed, which makes it possible to reveal a change in the structure of
attractors. The proposed method is tested on a model system of two
unidirectionally coupled logistic maps. It is shown that the method is robust
with respect to both the presence of a low-intensity noise and a nonlinear
distortion of the analyzed signal. Specific features of a rearranged structure
of the attractor of a driven subsystem in the example under consideration have
been studied. | nlin_CD |
A first-principles model of time-dependent variations in transmission
through a fluctuating scattering environment: Fading is the time-dependent variation in transmitted signal strength through
a complex medium, due to interference or temporally evolving multipath
scattering. In this paper we use random matrix theory (RMT) to establish a
first-principles model for fading, including both universal and non-universal
effects. This model provides a more general understanding of the most common
statistical models (Rayleigh fading and Rice fading) and provides a detailed
physical basis for their parameters. We also report experimental tests on two
ray-chaotic microwave cavities. The results show that our RMT model agrees with
the Rayleigh/Rice models in the high loss regime, but there are strong
deviations in low-loss systems where the RMT approach describes the data well. | nlin_CD |
Near action-degenerate periodic-orbit bunches: A skeleton of chaos: Long periodic orbits of hyperbolic dynamics do not exist as independent
individuals but rather come in closely packed bunches. Under weak resolution a
bunch looks like a single orbit in configuration space, but close inspection
reveals topological orbit-to-orbit differences. The construction principle of
bunches involves close self-"encounters" of an orbit wherein two or more
stretches stay close. A certain duality of encounters and the intervening
"links" reveals an infinite hierarchical structure of orbit bunches. -- The
orbit-to-orbit action differences $\Delta S$ within a bunch can be arbitrarily
small. Bunches with $\Delta S$ of the order of Planck's constant have
constructively interfering Feynman amplitudes for quantum observables, and this
is why the classical bunching phenomenon could yield the semiclassical
explanation of universal fluctuations in quantum spectra and transport. | nlin_CD |
Lie algebras in vortex dynamics and celestial mechanics - IV: The work of A.V. Borisov, A.E. Pavlov, Dynamics and Statics of Vortices on a
Plane and a Sphere - I (Reg. & Ch. Dynamics, 1998, Vol. 3, No 1, p.28-39)
introduces a naive description of dynamics of point vortices on a plane in
terms of variables of distances and areas which generate Lie-Poisson structure.
Using this approach a qualitative description of dynamics of point vortices on
a plane and a sphere is obtained in the works Dynamics of Three Vortices on a
Plane and a Sphere - II. General compact case by A.V. Borisov, V.G. Lebedev
(Reg. & Ch. Dynamics, 1998, Vol. 3, No 2, p.99-114), Dynamics of three vortices
on a plane and a sphere - III. Noncompact case. Problem of collaps and
scattering by A.V. Borisov, V.G. Lebedev (Reg. & Ch. Dynamics, 1998, Vol. 3, No
4, p.76-90). In this paper we consider more formal constructions of the general
problem of n vortices on a plane and a sphere. The developed methods of
algebraization are also applied to the classical problem of the reduction in
the three-body problem. | nlin_CD |
Wave chaos in the elastic disc: The relation between the elastic wave equation for plane, isotropic bodies
and an underlying classical ray dynamics is investigated. We study in
particular the eigenfrequencies of an elastic disc with free boundaries and
their connection to periodic rays inside the circular domain. Even though the
problem is separable, wave mixing between the shear and pressure component of
the wave field at the boundary leads to an effective stochastic part in the ray
dynamics. This introduces phenomena typically associated with classical chaos
as for example an exponential increase in the number of periodic orbits.
Classically, the problem can be decomposed into an integrable part and a simple
binary Markov process. Similarly, the wave equation can in the high frequency
limit be mapped onto a quantum graph. Implications of this result for the level
statistics are discussed. Furthermore, a periodic trace formula is derived from
the scattering matrix based on the inside-outside duality between eigen-modes
and scattering solutions and periodic orbits are identified by Fourier
transforming the spectral density | nlin_CD |
Asymptotic properties of the spectrum of neutral delay differential
equations: Spectral properties and transition to instability in neutral delay
differential equations are investigated in the limit of large delay. An
approximation of the upper boundary of stability is found and compared to an
analytically derived exact stability boundary. The approximate and exact
stability borders agree quite well for the large time delay, and the inclusion
of a time-delayed velocity feedback improves this agreement for small delays.
Theoretical results are complemented by a numerically computed spectrum of the
corresponding characteristic equations. | nlin_CD |
The hydrogen atom in an electric field: Closed-orbit theory with
bifurcating orbits: Closed-orbit theory provides a general approach to the semiclassical
description of photo-absorption spectra of arbitrary atoms in external fields,
the simplest of which is the hydrogen atom in an electric field. Yet, despite
its apparent simplicity, a semiclassical quantization of this system by means
of closed-orbit theory has not been achieved so far. It is the aim of this
paper to close that gap. We first present a detailed analytic study of the
closed classical orbits and their bifurcations. We then derive a simple form of
the uniform semiclassical approximation for the bifurcations that is suitable
for an inclusion into a closed-orbit summation. By means of a generalized
version of the semiclassical quantization by harmonic inversion, we succeed in
calculating high-quality semiclassical spectra for the hydrogen atom in an
electric field. | nlin_CD |
Concurrent formation of nearly synchronous clusters in each intertwined
cluster set with parameter mismatches: Cluster synchronization is a phenomenon in which oscillators in a given
network are partitioned into synchronous clusters. As recently shown, diverse
cluster synchronization patterns can be found using network symmetry when the
oscillators are identical. For such symmetry-induced cluster synchronization
patterns, subsets called intertwined clusters can exist, in which every cluster
in the same subset should synchronize or desynchronize concurrently. In this
work, to reflect the existence of noise in real systems, we consider networks
composed of nearly identical oscillators. We show that every cluster in the
same intertwined cluster set is nearly synchronized concurrently when the
nearly synchronous state of the set is stable. We also consider an extreme case
where only one cluster of an intertwined cluster set is composed of nearly
identical oscillators while every other cluster in the set is composed of
identical oscillators. In this case, deviation from the synchronous state of
every cluster in the same set increases linearly with the magnitude of
parameter mismatch within the cluster of nearly identical oscillators. We
confirm these results by numerical simulation. | nlin_CD |
Records and occupation time statistics for area-preserving maps: A relevant problem in dynamics is to characterize how deterministic systems
may exhibit features typically associated to stochastic processes. A widely
studied example is the study of (normal or anomalous) transport properties for
deterministic systems on a non-compact phase space. We consider here two
examples of area-preserving maps: the Chirikov-Taylor standard map and the
Casati-Prosen triangle map, and we investigate transport properties, records'
statistics and occupation time statistics. While the standard map, when a
chaotic sea is present, always reproduces results expected for simple random
walks, the triangle map -- whose analysis still displays many elusive points --
behaves in a wildly different way, some of the features being compatible with a
transient (non conservative) nature of the dynamics. | nlin_CD |
The topology of chaotic iterations: Chaotic iterations have been introduced on the one hand by Chazan, Mi- ranker
[6] and Miellou [10] in a numerical analysis context, and on the other hand by
Robert [12] and Pellegrin [11] in the discrete dynamical systems frame- work.
In both cases, the objective was to derive conditions of convergence of such
iterations to a fixed state. In this paper, a new point of view is presented,
the goal here is to derive conditions under which chaotic iterations admit a
chaotic behaviour in a rigorous mathematical sense. Contrary to what has been
studied in the literature, convergence is not desired.
More precisely, we establish in this paper a link between the concept of
chaotic iterations on a finite set and the notion of topological chaos [9],
[7], [8]. We are motivated by concrete applications of our approach, such as
the use of chaotic boolean iterations in the computer security field. Indeed,
the concept of chaos is used in many areas of data security without real
rigorous theoretical foundations, and without using the fundamental properties
that allow chaos. The wish of this paper is to bring a bit more mathematical
rigour in this field. This paper is an extension of[3], and a work in progress. | nlin_CD |
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