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Structure of the correlation function at the accumulation points of the
logistic map: The correlation function of the trajectory exactly at the Feigenbaum point of
the logistic map is investigated and checked by numerical experiments. Taking
advantage of recent closed analytical results on the symbol-to-symbol
correlation function of the generating partition, we are in position to justify
the deep algorithmic structure of the correlation function apart from numerical
constants. A generalization is given for arbitrary $m\cdot 2^{\infty}$
Feigenbaum attractors. | nlin_CD |
Convergence towards asymptotic state in 1-D mappings: a scaling
investigation: Decay to asymptotic steady state in one-dimensional logistic-like mappings is
characterized by considering a phenomenological description supported by
numerical simulations and confirmed by a theoretical description. As the
control parameter is varied bifurcations in the fixed points appear. We
verified at the bifurcation point in both; the transcritical, pitchfork and
period-doubling bifurcations, that the decay for the stationary point is
characterized via a homogeneous function with three critical exponents
depending on the nonlinearity of the mapping. Near the bifurcation the decay to
the fixed point is exponential with a relaxation time given by a power law
whose slope is independent of the nonlinearity. The formalism is general and
can be extended to other dissipative mappings. | nlin_CD |
Self-growing differential equations for hyperchaotic systems
reconstruction by modified genetic programming in a novel non-Lyapunov
approach: This paper has been withdrawn by the author due to a crucial sign error in
equation 1. | nlin_CD |
Sparse Identification of Slow Timescale Dynamics: Multiscale phenomena that evolve on multiple distinct timescales are
prevalent throughout the sciences. It is often the case that the governing
equations of the persistent and approximately periodic fast scales are
prescribed, while the emergent slow scale evolution is unknown. Yet the
course-grained, slow scale dynamics is often of greatest interest in practice.
In this work we present an accurate and efficient method for extracting the
slow timescale dynamics from signals exhibiting multiple timescales that are
amenable to averaging. The method relies on tracking the signal at
evenly-spaced intervals with length given by the period of the fast timescale,
which is discovered using clustering techniques in conjunction with the dynamic
mode decomposition. Sparse regression techniques are then used to discover a
mapping which describes iterations from one data point to the next. We show
that for sufficiently disparate timescales this discovered mapping can be used
to discover the continuous-time slow dynamics, thus providing a novel tool for
extracting dynamics on multiple timescales. | nlin_CD |
Classifying orbits in the classical Henon-Heiles Hamiltonian system: The H\'{e}non-Heiles potential is undoubtedly one of the most simple,
classical and characteristic Hamiltonian systems. The aim of this work is to
reveal the influence of the value of the total orbital energy, which is the
only parameter of the system, on the different families of orbits, by
monitoring how the percentage of chaotic orbits, as well as the percentages of
orbits composing the main regular families evolve when energy varies. In
particular, we conduct a thorough numerical investigation distinguishing
between ordered and chaotic orbits, considering only bounded motion for several
energy levels. The smaller alignment index (SALI) was computed by numerically
integrating the equations of motion as well as the variational equations to
extensive samples of orbits in order to distinguish safely between ordered and
chaotic motion. In addition, a method based on the concept of spectral dynamics
that utilizes the Fourier transform of the time series of each coordinate is
used to identify the various families of regular orbits and also to recognize
the secondary resonances that bifurcate from them. Our exploration takes place
both in the physical $(x,y)$ and the phase $(y,\dot{y})$ space for a better
understanding of the orbital properties of the system. It was found, that for
low energy levels the motion is entirely regular being the box orbits the most
populated family, while as the value of the energy increases chaos and several
resonant families appear. We also observed, that the vast majority of the
resonant orbits belong in fact in bifurcated families of the main 1:1 resonant
family. We have also compared our results with previous similar outcomes
obtained using different chaos indicators. | nlin_CD |
An approach to chaotic synchronization: This paper deals with the chaotic oscillator synchronization. A new approach
to the synchronization of chaotic oscillators has been proposed. This approach
is based on the analysis of different time scales in the time series generated
by the coupled chaotic oscillators. It has been shown that complete
synchronization, phase synchronization, lag synchronization and generalized
synchronization are the particular cases of the synchronized behavior called as
"time-scale synchronization". The quantitative measure of chaotic oscillator
synchronous behavior has been proposed. This approach has been applied for the
coupled R\"ossler systems and two coupled Chua's circuits. | nlin_CD |
A method to determine structural patterns of mechanical systems with
impacts: A structural classification method of vibro-impact systems with an arbitrary
finite number of degrees of freedom based on the principles given by
Blazejczyk-Okolewska et al. [Blazejczyk- Okolewska B., Czolczynski K.,
Kapitaniak T., Classification principles of types of mechanical systems with
impacts - fundamental assumptions and rules, European Journal of Mechanics
A/Solids, 2004, 23, pp. 517-537] has been proposed. We provide a
characterization of equivalent mechanical systems with impacts expressed in
terms of a new matrix representation, introduced to formulate the notation of
the relations occurring in the system. The developed identification and
elimination procedures of equivalent systems and an identification procedure of
connected systems enable determination of a set of all structural patterns of
vibro-impact systems with an arbitrary finite number of degrees of freedom. | nlin_CD |
Heat conduction and Fourier's law in a class of many particle dispersing
billiards: We consider the motion of many confined billiard balls in interaction and
discuss their transport and chaotic properties. In spite of the absence of mass
transport, due to confinement, energy transport can take place through binary
collisions between neighbouring particles. We explore the conditions under
which relaxation to local equilibrium occurs on time scales much shorter than
that of binary collisions, which characterize the transport of energy, and
subsequent relaxation to local thermal equilibrium. Starting from the
pseudo-Liouville equation for the time evolution of phase-space distributions,
we derive a master equation which governs the energy exchange between the
system constituents. We thus obtain analytical results relating the transport
coefficient of thermal conductivity to the frequency of collision events and
compute these quantities. We also provide estimates of the Lyapunov exponents
and Kolmogorov-Sinai entropy under the assumption of scale separation. The
validity of our results is confirmed by extensive numerical studies. | nlin_CD |
Reducing or enhancing chaos using periodic orbits: A method to reduce or enhance chaos in Hamiltonian flows with two degrees of
freedom is discussed. This method is based on finding a suitable perturbation
of the system such that the stability of a set of periodic orbits changes
(local bifurcations). Depending on the values of the residues, reflecting their
linear stability properties, a set of invariant tori is destroyed or created in
the neighborhood of the chosen periodic orbits. An application on a
paradigmatic system, a forced pendulum, illustrates the method. | nlin_CD |
Marginal resonances and intermittent behaviour in the motion in the
vicinity of a separatrix: A condition upon which sporadic bursts (intermittent behaviour) of the
relative energy become possible is derived for the motion in the chaotic layer
around the separatrix of non-linear resonance. This is a condition for the
existence of a marginal resonance, i.e. a resonance located at the border of
the layer. A separatrix map in Chirikov's form [Chirikov, B. V., Phys. Reports
52, 263 (1979)] is used to describe the motion. In order to provide a
straightforward comparison with numeric integrations, the separatrix map is
synchronized to the surface of the section farthest from the saddle point. The
condition of intermittency is applied to clear out the nature of the phenomenon
of bursts of the eccentricity of chaotic asteroidal trajectories in the 3/1
mean motion commensurability with Jupiter. On the basis of the condition, a new
intermittent regime of resonant asteroidal motion is predicted and then
identified in numeric simulations. | nlin_CD |
Transition from amplitude to oscillation death in a network of
oscillators: We report a transition from a homogeneous steady state (HSS) to inhomogeneous
steady states (IHSSs) in a network of globally coupled identical oscillators.
We perturb a synchronized population in the network with a few local negative
mean field links. It is observed that the whole population splits into two
clusters for a certain number of negative mean field links and specific range
of coupling strength. For further increases of the strength of interaction
these clusters collapse to a HSS followed by a transition to IHSSs. We
analytically determine the origin of HSS and its transition to IHSS in relation
to the number of negative mean-field links and the strength of interaction
using a reductionism approach to the model network in a two-cluster state. We
verify the results with numerical examples of networks using the paradigmatic
Landau-Stuart limit cycle system and the chaotic Rossler oscillator as
dynamical nodes. During the transition from HSS to IHSSs, the network follows
the Turing type symmetry breaking pitchfork or transcritical bifurcation
depending upon the system dynamics. | nlin_CD |
Short wave length approximation of a boundary integral operator for
homogeneous and isotropic elastic bodies: We derive a short wave length approximation of a boundary integral operator
for two-dimensional isotropic and homogeneous elastic bodies of arbitrary
shape. Trace formulae for elastodynamics can be deduced in this way from first
principles starting directly from Navier-Cauchy's equation. | nlin_CD |
Families of piecewise linear maps with constant Lyapunov exponent: We consider families of piecewise linear maps in which the moduli of the two
slopes take different values. In some parameter regions, despite the variations
in the dynamics, the Lyapunov exponent and the topological entropy remain
constant. We provide numerical evidence of this fact and we prove it
analytically for some special cases. The mechanism is very different from that
of the logistic map and we conjecture that the Lyapunov plateaus reflect
arithmetic relations between the slopes. | nlin_CD |
What Are the New Implications of Chaos for Unpredictability?: From the beginning of chaos research until today, the unpredictability of
chaos has been a central theme. It is widely believed and claimed by
philosophers, mathematicians and physicists alike that chaos has a new
implication for unpredictability, meaning that chaotic systems are
unpredictable in a way that other deterministic systems are not. Hence one
might expect that the question 'What are the new implications of chaos for
unpredictability?' has already been answered in a satisfactory way. However,
this is not the case. I will critically evaluate the existing answers and argue
that they do not fit the bill. Then I will approach this question by showing
that chaos can be defined via mixing, which has not been explicitly argued for.
Based on this insight, I will propose that the sought-after new implication of
chaos for unpredictability is the following: for predicting any event all
sufficiently past events are approximately probabilistically irrelevant. | nlin_CD |
New Scenario to Chaos Transition in the Mappings with Discontinuities: We consider a many-parametric piecewise mapping with discontinuity. That is a
one dimensional model of singular dynamic system. The stability boundary are
calculated analytically and numerically. New typical features of stable cycle
structures and scenario to chaos transition provoked by discontinuity are
found. | nlin_CD |
Integrating Random Matrix Theory Predictions with Short-Time Dynamical
Effects in Chaotic Systems: We discuss a modification to Random Matrix Theory eigenstate statistics, that
systematically takes into account the non-universal short-time behavior of
chaotic systems. The method avoids diagonalization of the Hamiltonian, instead
requiring only a knowledge of short-time dynamics for a chaotic system or
ensemble of similar systems. Standard Random Matrix Theory and semiclassical
predictions are recovered in the limits of zero Ehrenfest time and infinite
Heisenberg time, respectively. As examples, we discuss wave function
autocorrelations and cross-correlations, and show that significant improvement
in accuracy is obtained for simple chaotic systems where comparison can be made
with brute-force diagonalization. The accuracy of the method persists even when
the short-time dynamics of the system or ensemble is known only in a classical
approximation. Further improvement in the rate of convergence is obtained when
the method is combined with the correlation function bootstrapping approach
introduced previously. | nlin_CD |
A wavelet-based tool for studying non-periodicity: This paper presents a new numerical approach to the study of non-periodicity
in signals, which can complement the maximal Lyapunov exponent method for
determining chaos transitions of a given dynamical system. The proposed
technique is based on the continuous wavelet transform and the wavelet
multiresolution analysis. A new parameter, the \textit{scale index}, is
introduced and interpreted as a measure of the degree of the signal's
non-periodicity. This methodology is successfully applied to three classical
dynamical systems: the Bonhoeffer-van der Pol oscillator, the logistic map, and
the Henon map. | nlin_CD |
Anti-Synchronization in Multiple Time Delay Power Systems: We investigate chaos antisynchronization between two uni-directionally
coupled multiple time delay power systems.The results are of certain importance
to prevent power black-out in the entire power grid. | nlin_CD |
Fractional dynamics of systems with long-range interaction: We consider one-dimensional chain of coupled linear and nonlinear oscillators
with long-range power wise interaction defined by a term proportional to
1/|n-m|^{\alpha+1}. Continuous medium equation for this system can be obtained
in the so-called infrared limit when the wave number tends to zero. We
construct a transform operator that maps the system of large number of ordinary
differential equations of motion of the particles into a partial differential
equation with the Riesz fractional derivative of order \alpha, when 0<\alpha<2.
Few models of coupled oscillators are considered and their synchronized states
and localized structures are discussed in details. Particularly, we discuss
some solutions of time-dependent fractional Ginzburg-Landau (or nonlinear
Schrodinger) equation. | nlin_CD |
Chaotic switching in driven-dissipative Bose-Hubbard dimers: when a flip
bifurcation meets a T-point in $R^4$: The Bose--Hubbard dimer model is a celebrated fundamental quantum mechanical
model that accounts for the dynamics of bosons at two interacting sites. It has
been realized experimentally by two coupled, driven and lossy photonic crystal
nanocavities, which are optical devices that operate with only a few hundred
photons due to their extremely small size. Our work focuses on characterizing
the different dynamics that arise in the semiclassical approximation of such
driven-dissipative photonic Bose--Hubbard dimers. Mathematically, this system
is a four-dimensional autonomous vector field that describes two specific
coupled oscillators, where both the amplitude and the phase are important. We
perform a bifurcation analysis of this system to identify regions of different
behavior as the pump power $f$ and the detuning $\delta$ of the driving signal
are varied, for the case of fixed positive coupling. The bifurcation diagram in
the $(f,\delta)$-plane is organized by two points of codimension-two
bifurcations -- a $Z_2$-equivariant homoclinic flip bifurcation and a Bykov
T-point -- and provides a roadmap for the observable dynamics, including
different types of chaotic behavior. To illustrate the overall structure and
different accumulation processes of bifurcation curves and associated regions,
our bifurcation analysis is complemented by the computation of kneading
invariants and of maximum Lyapunov exponents in the $(f,\delta)$-plane. The
bifurcation diagram displays a menagerie of dynamical behavior and offers
insights into the theory of global bifurcations in a four-dimensional phase
space, including novel bifurcation phenomena such as degenerate singular
heteroclinic cycles. | nlin_CD |
2D and 3D Dense-Fluid Shear Flows via Nonequilibrium Molecular Dynamics.
Comparison of Time-and-Space-Averaged Tensor Temperature and Normal Stresses
from Doll's, Sllod, and Boundary-Driven Shear Algorithms: Homogeneous shear flows (with constant strainrate du/dy) are generated with
the Doll's and Sllod algorithms and compared to corresponding inhomogeneous
boundary-driven flows. We use one-, two-, and three-dimensional smooth-particle
weight functions for computing instantaneous spatial averages. The nonlinear
stress differences are small, but significant, in both two and three space
dimensions. In homogeneous systems the sign and magnitude of the shearplane
stress difference, P(xx) - P(yy), depend on both the thermostat type and the
chosen shearflow algorithm. The Doll's and Sllod algorithms predict opposite
signs for this stress difference, with the Sllod approach definitely wrong, but
somewhat closer to the (boundary-driven) truth. Neither of the homogeneous
shear algorithms predicts the correct ordering of the kinetic temperatures,
T(xx) > T(zz) > T(yy). | nlin_CD |
Twenty-five years of multifractals in fully developed turbulence: a
tribute to Giovanni Paladin: The paper {\it On the multifractal nature of fully developed turbulence and
chaotic systems}, by R. Benzi {\it et al.} published in this journal in 1984
(vol {\bf 17}, page 3521) has been a starting point of many investigations on
the different faces of selfsimilarity and intermittency in turbulent phenomena.
Since then, the multifractal model has become a useful tool for the study of
small scale turbulence, in particular for detailed predictions of different
Eulerian and Lagrangian statistical properties. In the occasion of the 50-th
birthday of our unforgettable friend and colleague Giovanni Paladin
(1958-1996), we review here the basic concepts and some applications of the
multifractal model for turbulence. | nlin_CD |
Elastic turbulence in a polymer solution flow: Turbulence is one of the most fascinating phenomena in nature and one of the
biggest challenges for modern physics. It is common knowledge that a flow of a
simple, Newtonian fluid is likely to be turbulent, when velocity is high,
viscosity is low and size of the tank is large\cite{Landau,Tritt}. Solutions of
flexible long-chain polymers are known as visco-elastic fluids\cite{bird}. In
our experiments we show, that flow of a polymer solution with large enough
elasticity can become quite irregular even at low velocity, high viscosity and
in a small tank. The fluid motion is excited in a broad range of spatial and
temporal scales. The flow resistance increases by a factor of about twenty. So,
while the Reynolds number, $\boldmath{Re}$, may be arbitrary low, the observed
flow has all main features of developed turbulence, and can be compared to
turbulent flow in a pipe at $\bf {Re\simeq 10^5}$\cite{Landau,Tritt}. This {\it
elastic turbulence} is accompanied by significant stretching of the polymer
molecules, and the resulting increase of the elastic stresses can reach two
orders of magnitude. | nlin_CD |
A note on dissipation in helical turbulence: In helical turbulence a linear cascade of helicity accompanying the energy
cascade has been suggested. Since energy and helicity have different
dimensionality we suggest the existence of a characteristic inner scale,
$\xi=k_H^{-1}$, for helicity dissipation in a regime of hydrodynamic fully
developed turbulence and estimate it on dimensional grounds. This scale is
always larger than the Kolmogorov scale, $\eta=k_E^{-1}$, and their ratio $\eta
/ \xi $ vanishes in the high Reynolds number limit, so the flow will always be
helicity free in the small scales. | nlin_CD |
Universal behaviour of a wave chaos based electromagnetic reverberation
chamber: In this article, we present a numerical investigation of three-dimensional
electromagnetic Sinai-like cavities. We computed around 600 eigenmodes for two
different geometries: a parallelepipedic cavity with one half- sphere on one
wall and a parallelepipedic cavity with one half-sphere and two spherical caps
on three adjacent walls. We show that the statistical requirements of a well
operating reverberation chamber are better satisfied in the more complex
geometry without a mechanical mode-stirrer/tuner. This is to the fact that our
proposed cavities exhibit spatial and spectral statistical behaviours very
close to those predicted by random matrix theory. More specifically, we show
that in the range of frequency corresponding to the first few hundred modes,
the suppression of non-generic modes (regarding their spatial statistics) can
be achieved by reducing drastically the amount of parallel walls. Finally, we
compare the influence of losses on the statistical complex response of the
field inside a parallelepipedic and a chaotic cavity. We demonstrate that, in a
chaotic cavity without any stirring process, the low frequency limit of a well
operating reverberation chamber can be significantly reduced under the usual
values obtained in mode-stirred reverberation chambers. | nlin_CD |
Resonance states of the three-disk scattering system: For the paradigmatic three-disk scattering system, we confirm a recent
conjecture for open chaotic systems, which claims that resonance states are
composed of two factors. In particular, we demonstrate that one factor is given
by universal exponentially distributed intensity fluctuations. The other
factor, supposed to be a classical density depending on the lifetime of the
resonance state, is found to be very well described by a classical
construction. Furthermore, ray-segment scars, recently observed in dielectric
cavities, dominate every resonance state at small wavelengths also in the
three-disk scattering system. We introduce a new numerical method for computing
resonances, which allows for going much further into the semiclassical limit.
As a consequence we are able to confirm the fractal Weyl law over a
correspondingly large range. | nlin_CD |
The Decay of Passive Scalars Under the Action of Single Scale Smooth
Velocity Fields in Bounded 2D Domains : From non self similar pdf's to self
similar eigenmodes: We examine the decay of passive scalars with small, but non zero, diffusivity
in bounded 2D domains. The velocity fields responsible for advection are smooth
(i.e., they have bounded gradients) and of a single large scale. Moreover, the
scale of the velocity field is taken to be similar to the size of the entire
domain. The importance of the initial scale of variation of the scalar field
with respect to that of the velocity field is strongly emphasized. If these
scales are comparable and the velocity field is time periodic, we see the
formation of a periodic scalar eigenmode. The eigenmode is numerically realized
by means of a deterministic 2D map on a lattice. Analytical justification for
the eigenmode is available from theorems in the dynamo literature. Weakening
the notion of an eigenmode to mean statistical stationarity, we provide
numerical evidence that the eigenmode solution also holds for aperiodic flows
(represented by random maps). Turning to the evolution of an initially small
scale scalar field, we demonstrate the transition from an evolving (i.e., {\it
non} self similar) pdf to a stationary (self similar) pdf as the scale of
variation of the scalar field progresses from being small to being comparable
to that of the velocity field (and of the domain). Furthermore, the {\it non}
self similar regime itself consists of two stages. Both the stages are examined
and the coupling between diffusion and the distribution of the Finite Time
Lyapunov Exponents is shown to be responsible for the pdf evolution. | nlin_CD |
Intermittency and Synchronisation in Gumowski-Mira Maps: The Gumowski-Mira map is a 2-dimensional recurrence relation that provide a
large variety of phase space plots resembling fractal patterns of nature. We
investigate the nature of the dynamical states that produce these patterns and
find that they correspond to Type I intermittency near periodic cycles. By
coupling two GM maps, such patterns can be wiped out to give synchronised
periodic states of lower order.The efficiency of the coupling scheme is
established by analysing the error function dynamics. | nlin_CD |
A self-synchronizing stream cipher based on chaotic coupled maps: A revised self-synchronizing stream cipher based on chaotic coupled maps is
proposed. This system adds input and output functions aim to strengthen its
security. The system performs basic floating-point analytical computation on
real numbers, incorporating auxiliarily with algebraic operations on integer
numbers. | nlin_CD |
The overlapping of nonlinear resonances and problem of quantum chaos: The motion of a nonlinearly oscilating partical under the influence of a
periodic sequence of short impulses is investigated. We analyze the Schrodinger
equation for the universal Hamiltonian. The idea about the emerging of quantum
chaos due to the adiabatic motion along the curves of Mathieu characteristics
at multiple passages through the points of branching is advanced | nlin_CD |
The Dynamical Matching Mechanism in Phase Space for Caldera-Type
Potential Energy Surfaces: Dynamical matching occurs in a variety of important organic chemical
reactions. It is observed to be a result of a potential energy surface (PES)
having specific geometric features. In particular, a region of relative
flatness where entrance and exit to this region is controlled by index-one
saddles. Examples of potential energy surfaces having these features are the
so-called caldera potential energy surfaces. We develop a predictive level of
understanding of the phenomenon of dynamical matching in a caldera potential
energy surface. We show that the phase space structure that governs dynamical
matching is a particular type of heteroclinic trajectory which gives rise to
trapping of trajectories in the central region of the caldera PES. When the
heteroclinic trajectory is broken, as a result of parameter variations, then
dynamical matching occurs. | nlin_CD |
The oscillating two-cluster chimera state in non-locally coupled phase
oscillators: We investigate an array of identical phase oscillators non-locally coupled
without time delay, and find that chimera state with two coherent clusters
exists which is only reported in delay-coupled systems previously. Moreover, we
find that the chimera state is not stationary for any finite number of
oscillators. The existence of the two-cluster chimera state and its
time-dependent behaviors for finite number of oscillators are confirmed by the
theoretical analysis based on the self-consistency treatment and the
Ott-Antonsen ansatz. | nlin_CD |
Experimental test of a trace formula for two-dimensional dielectric
resonators: Resonance spectra of two-dimensional dielectric microwave resonators of
circular and square shapes have been measured. The deduced length spectra of
periodic orbits were analyzed and a trace formula for dielectric resonators
recently proposed by Bogomolny et al. [Phys. Rev. E 78, 056202 (2008)] was
tested. The observed deviations between the experimental length spectra and the
predictions of the trace formula are attributed to a large number of missing
resonances in the measured spectra. We show that by taking into account the
systematics of observed and missing resonances the experimental length spectra
are fully understood. In particular, a connection between the most long-lived
resonances and certain periodic orbits is established experimentally. | nlin_CD |
Dynamical Analysis of a Networked Control System: A new network data transmission strategy was proposed in Zhang \& Chen [2005]
(arXiv:1405.2404), where the resulting nonlinear system was analyzed and the
effectiveness of the transmission strategy was demonstrated via simulations. In
this paper, we further generalize the results of Zhang \& Chen [2005] in the
following ways: 1) Construct first-return maps of the nonlinear systems
formulated in Zhang \& Chen [2005] and derive several existence conditions of
periodic orbits and study their properties. 2) Formulate the new system as a
hybrid system, which will ease the succeeding analysis. 3) Prove that this type
of hybrid systems is not structurally stable based on phase transition which
can be applied to higher-dimensional cases effortlessly. 4) Simulate a
higher-dimensional model with emphasis on their rich dynamics. 5) Study a class
of continuous-time hybrid systems as the counterparts of the discrete-time
systems discussed above. 6) Propose new controller design methods based on this
network data transmission strategy to improve the performance of each
individual system and the whole network. We hope that this research and the
problems posed here will rouse interests of researchers in such fields as
control, dynamical systems and numerical analysis. | nlin_CD |
Influence of weak anisotropy on scaling regimes in a model of advected
vector field: Influence of weak uniaxial small-scale anisotropy on the stability of
inertial-range scaling regimes in a model of a passive transverse vector field
advected by an incompressible turbulent flow is investigated by means of the
field theoretic renormalization group. Weak anisotropy means that parameters
which describe anisotropy are chosen to be close to zero, therefore in all
expressions it is enough to leave only linear terms in anisotropy parameters.
Turbulent fluctuations of the velocity field are taken to have the Gaussian
statistics with zero mean and defined noise with finite correlations in time.
It is shown that stability of the inertial-range scaling regimes in the
three-dimensional case is not destroyed by anisotropy but the corresponding
stability of the two-dimensional system can be destroyed even by the presence
of weak anisotropy. A borderline dimension $d_c$ below which the stability of
the scaling regime is not present is calculated as a function of anisotropy
parameters. | nlin_CD |
Peeling Bifurcations of Toroidal Chaotic Attractors: Chaotic attractors with toroidal topology (van der Pol attractor) have
counterparts with symmetry that exhibit unfamiliar phenomena. We investigate
double covers of toroidal attractors, discuss changes in their morphology under
correlated peeling bifurcations, describe their topological structures and the
changes undergone as a symmetry axis crosses the original attractor, and
indicate how the symbol name of a trajectory in the original lifts to one in
the cover. Covering orbits are described using a powerful synthesis of kneading
theory with refinements of the circle map. These methods are applied to a
simple version of the van der Pol oscillator. | nlin_CD |
Low-frequency regime transitions and predictability of regimes in a
barotropic model: Predictability of flow is examined in a barotropic vorticity model that
admits low frequency regime transitions between zonal and dipolar states. Such
transitions in the model were first studied by Bouchet and Simonnet (2009) and
are reminiscent of regime change phenomena in the weather and climate systems
wherein extreme and abrupt qualitative changes occur, seemingly randomly, after
long periods of apparent stability. Mechanisms underlying regime transitions in
the model are not well understood yet. From the point of view of atmospheric
and oceanic dynamics, a novel aspect of the model is the lack of any source of
background gradient of potential-vorticity such as topography or planetary
gradient of rotation rate (e.g., as in Charney & DeVore '79).
We consider perturbations that are embedded onto the system's chaotic
attractor under the full nonlinear dynamics as bred vectors---nonlinear
generalizations of the leading (backward) Lyapunov vector. We find that
ensemble predictions that use bred vector perturbations are more robust in
terms of error-spread relationship than those that use Lyapunov vector
perturbations. In particular, when bred vector perturbations are used in
conjunction with a simple data assimilation scheme (nudging to truth), we find
that at least some of the evolved perturbations align to identify
low-dimensional subspaces associated with regions of large forecast error in
the control (unperturbed, data-assimilating) run; this happens less often in
ensemble predictions that use Lyapunov vector perturbations. Nevertheless, in
the inertial regime we consider, we find that (a) the system is more
predictable when it is in the zonal regime, and that (b) the horizon of
predictability is far too short compared to characteristic time scales
associated with processes that lead to regime transitions, thus precluding the
possibility of predicting such transitions. | nlin_CD |
Extended Harmonic Map Equations and the Chaotic Soliton Solutions: In this paper, the theory of harmonic maps is extended. The soliton or
traveling wave solutions of Euler's equations of the extended harmonic maps are
studied. In certain cases, the chaotic behaviors of these partial equations can
be found for the particular case of the metrics and the potential functions of
the extended harmonic equations. | nlin_CD |
Standard map-like models for single and multiple walkers in an annular
cavity: Recent experiments on walking droplets in an annular cavity showed the
existence of complex dynamics including chaotically changing velocity. This
article presents models, influenced by the kicked rotator/standard map, for
both single and multiple droplets. The models are shown to achieve both
qualitative and quantitative agreement with the experiments, and makes
predictions about heretofore unobserved behavior. Using dynamical systems
techniques and bifurcation theory, the single droplet model is analyzed to
prove dynamics suggested by the numerical simulations. | nlin_CD |
Non-integrability of restricted double pendula: We consider two special types of double pendula, with the motion of masses
restricted to various surfaces. In order to get quick insight into the dynamics
of the considered systems the Poincar\'e cross sections as well as bifurcation
diagrams have been used. The numerical computations show that both models are
chaotic which suggest that they are not integrable. We give an analytic proof
of this fact checking the properties of the differential Galois group of the
system's variational equations along a particular non-equilibrium solution. | nlin_CD |
Perturbation-Free Prediction of Resonance-Assisted Tunneling in Mixed
Regular--Chaotic Systems: For generic Hamiltonian systems we derive predictions for dynamical tunneling
from regular to chaotic phase-space regions. In contrast to previous
approaches, we account for the resonance-assisted enhancement of
regular-to-chaotic tunneling in a non-perturbative way. This provides the
foundation for future semiclassical complex-path evaluations of
resonance-assisted regular-to-chaotic tunneling. Our approach is based on a new
class of integrable approximations which mimic the regular phase-space region
and its dominant nonlinear resonance chain in a mixed regular--chaotic system.
We illustrate the method for the standard map. | nlin_CD |
Echoes in classical dynamical systems: Echoes arise when external manipulations to a system induce a reversal of its
time evolution that leads to a more or less perfect recovery of the initial
state. We discuss the accuracy with which a cloud of trajectories returns to
the initial state in classical dynamical systems that are exposed to additive
noise and small differences in the equations of motion for forward and backward
evolution. The cases of integrable and chaotic motion and small or large noise
are studied in some detail and many different dynamical laws are identified.
Experimental tests in 2-d flows that show chaotic advection are proposed. | nlin_CD |
Rattling and freezing in a 1-D transport model: We consider a heat conduction model introduced in \cite{Collet-Eckmann 2009}.
This is an open system in which particles exchange momentum with a row of
(fixed) scatterers. We assume simplified bath conditions throughout, and give a
qualitative description of the dynamics extrapolating from the case of a single
particle for which we have a fairly clear understanding. The main phenomenon
discussed is {\it freezing}, or the slowing down of particles with time. As
particle number is conserved, this means fewer collisions per unit time, and
less contact with the baths; in other words, the conductor becomes less
effective. Careful numerical documentation of freezing is provided, and a
theoretical explanation is proposed. Freezing being an extremely slow process,
however, the system behaves as though it is in a steady state for long
durations. Quantities such as energy and fluxes are studied, and are found to
have curious relationships with particle density. | nlin_CD |
On nonlinear fractional maps: Nonlinear maps with power-law memory: This article is a short review of the recent results on properties of
nonlinear fractional maps which are maps with power- or asymptotically
power-law memory. These maps demonstrate the new type of attractors - cascade
of bifurcations type trajectories, power-law convergence/divergence of
trajectories, period doubling bifurcations with changes in the memory
parameter, intersection of trajectories, and overlapping of attractors. In the
limit of small time steps these maps converge to nonlinear fractional
differential equations. | nlin_CD |
Intermittent Peel Front Dynamics and the Crackling Noise in an Adhesive
Tape: We report a comprehensive investigation of a model for peeling of an adhesive
tape along with a nonlinear time series analysis of experimental acoustic
emission signals in an effort to understand the origin of intermittent peeling
of an adhesive tape and its connection to acoustic emission. The model
represents the acoustic energy dissipated in terms of Rayleigh dissipation
functional that depends on the local strain rate. We show that the nature of
the peel front exhibits rich spatiotemporal patterns ranging from smooth,
rugged and stuck-peeled configurations that depend on three parameters, namely,
the ratio of inertial time scale of the tape mass to that of the roller, the
dissipation coefficient and the pull velocity. The stuck-peeled configurations
are reminiscent of fibrillar peel front patterns observed in experiments. We
show that while the intermittent peeling is controlled by the peel force
function, the model acoustic energy dissipated depends on the nature of the
peel front and its dynamical evolution. Even though the acoustic energy is a
fully dynamical quantity, it can be quite noisy for a certain set of parameter
values suggesting the deterministic origin of acoustic emission in experiments.
To verify this suggestion, we have carried out a dynamical analysis of
experimental acoustic emission time series for a wide range of traction
velocities. Our analysis shows an unambiguous presence of chaotic dynamics
within a subinterval of pull speeds within the intermittent regime. Time series
analysis of the model acoustic energy signals is also found to be chaotic
within a subinterval of pull speeds. | nlin_CD |
Explosive synchronization transition in a ring of coupled oscillators: Explosive synchronization(ES), as one kind of abrupt dynamical transition in
nonlinearly coupled systems, is currently a subject of great interests. Given a
special frequency distribution, a mixed ES is observed in a ring of coupled
phase oscillators which transit from partial synchronization to ES with the
increment of coupling strength. The coupling weight is found to control the
size of the hysteresis region where asynchronous and synchronized states
coexist. Theoretical analysis reveals that the transition varies from the mixed
ES, to the ES and then to a continuous one with increasing coupling weight. Our
results are helpful to extend the understanding of the ES in homogenous
networks. | nlin_CD |
Multi-locality and fusion rules on the generalized structure functions
in two-dimensional and three-dimensional Navier-Stokes turbulence: Using the fusion rules hypothesis for three-dimensional and two-dimensional
Navier-Stokes turbulence, we generalize a previous non-perturbative locality
proof to multiple applications of the nonlinear interactions operator on
generalized structure functions of velocity differences. We shall call this
generalization of non-perturbative locality to multiple applications of the
nonlinear interactions operator "multilocality". The resulting cross-terms pose
a new challenge requiring a new argument and the introduction of a new fusion
rule that takes advantage of rotational symmetry. Our main result is that the
fusion rules hypothesis implies both locality and multilocality in both the IR
and UV limits for the downscale energy cascade of three-dimensional
Navier-Stokes turbulence and the downscale enstrophy cascade and inverse energy
cascade of two-dimensional Navier-Stokes turbulence. We stress that these
claims relate to non-perturbative locality of generalized structure functions
on all orders, and not the term by term perturbative locality of diagrammatic
theories or closure models that involve only two-point correlation and response
functions. | nlin_CD |
Dynamics of Three Non-co-rotating Vortices in Bose-Einstein Condensates: In this work we use standard Hamiltonian-system techniques in order to study
the dynamics of three vortices with alternating charges in a confined
Bose-Einstein condensate. In addition to being motivated by recent experiments,
this system offers a natural vehicle for the exploration of the transition of
the vortex dynamics from ordered to progressively chaotic behavior. In
particular, it possesses two integrals of motion, the {\it energy} (which is
expressed through the Hamiltonian $H$) and the {\it angular momentum} $L$ of
the system. By using the integral of the angular momentum, we reduce the system
to a two degree-of-freedom one with $L$ as a parameter and reveal the topology
of the phase space through the method of Poincar\'e surfaces of section.
We categorize the various motions that appear in the different regions of the
sections and we study the major bifurcations that occur to the families of
periodic motions of the system. Finally, we correspond the orbits on the
surfaces of section to the real space motion of the vortices in the plane. | nlin_CD |
Chaotic mechanism description by an elementary mixer for the template of
an attractor: Templates can be used to describe the topological properties of chaotic
attractors. For attractors bounded by genus one torus, these templates are
described by a linking matrix. For a given attractor, it has been shown that
the template depends on the Poincar\'e section chosen to performed the
analysis. The purpose of this article is to present an algorithm that gives the
elementary mixer of a template in order to have a unique way to describe a
chaotic mechanism. This chaotic mechanism is described with a linking matrix
and we also provide a method to generate and classify all the possible chaotic
mechanisms made of two to five strips. | nlin_CD |
Chimera-Like Coexistence of Synchronized Oscillation and Death in an
Ecological Network: We report a novel spatiotemporal state, namely the chimera-like incongruous
coexistence of {\it synchronized oscillation} and {\it stable steady state}
(CSOD) in a realistic ecological network of nonlocally coupled oscillators.
Unlike the {\it chimera} and {\it chimera death} state, in the CSOD state
identical oscillators are self-organized into two coexisting spatially
separated domains: In one domain neighboring oscillators show synchronized
oscillation and in another domain the neighboring oscillators randomly populate
either a synchronized oscillating state or a stable steady state (we call it a
death state). We show that the interplay of nonlocality and coupling strength
results in two routes to the CSOD state: One is from a coexisting mixed state
of amplitude chimera and death, and another one is from a globally synchronized
state. We further explore the importance of this study in ecology that gives a
new insight into the relationship between spatial synchrony and global
extinction of species. | nlin_CD |
Chaos on a High-Dimensional Torus: Transition from quasiperiodicity with many frequencies (i.e., a
high-dimensional torus) to chaos is studied by using $N$-dimensional globally
coupled circle maps. First, the existence of $N$-dimensional tori with $N\geq
2$ is confirmed while they become exponentially rare with $N$. Besides, chaos
exists even when the map is invertible, and such chaos has more null Lyapunov
exponents as $N$ increases. This unusual form of "chaos on a torus," termed
toric chaos, exhibits delocalization and slow dynamics of the first Lyapunov
vector. Fractalization of tori at the transition to chaos is also suggested.
The relevance of toric chaos to neural dynamics and turbulence is discussed in
relation to chaotic itinerancy. | nlin_CD |
Relaxation of finite perturbations: Beyond the Fluctuation-Response
relation: We study the response of dynamical systems to finite amplitude perturbation.
A generalized Fluctuation-Response relation is derived, which links the average
relaxation toward equilibrium to the invariant measure of the system and points
out the relevance of the amplitude of the initial perturbation. Numerical
computations on systems with many characteristic times show the relevance of
the above relation in realistic cases. | nlin_CD |
Assessing the direction of climate interactions by means of complex
networks and information theoretic tools: An estimate of the net direction of climate interactions in different
geographical regions is made by constructing a directed climate network from a
regular latitude-longitude grid of nodes, using a directionality index (DI)
based on conditional mutual information. Two datasets of surface air
temperature anomalies - one monthly-averaged and another daily-averaged - are
analyzed and compared. The network links are interpreted in terms of known
atmospheric tropical and extratropical variability patterns. Specific and
relevant geographical regions are selected, the net direction of propagation of
the atmospheric patterns is analyzed and the direction of the inferred links is
validated by recovering some well-known climate variability structures. These
patterns are found to be acting at various time-scales, such as atmospheric
waves in the extra-tropics or longer range events in the tropics. This analysis
demonstrates the capability of the DI measure to infer the net direction of
climate interactions and may contribute to improve the present understanding of
climate phenomena and climate predictability. The work presented here also
stands out as an application of advanced tools to the analysis of empirical,
real-world data. | nlin_CD |
Experimental observation of a complex periodic window: The existence of a special periodic window in the two-dimensional parameter
space of an experimental Chua's circuit is reported. One of the main reasons
that makes such a window special is that the observation of one implies that
other similar periodic windows must exist for other parameter values. However,
such a window has never been experimentally observed, since its size in
parameter space decreases exponentially with the period of the periodic
attractor. This property imposes clear limitations for its experimental
detection. | nlin_CD |
Estimating short-time period to break different types of chaotic
modulation based secure communications: In recent years, chaotic attractors have been extensively used in the design
of secure communication systems. One of the preferred ways of transmitting the
information signal is binary chaotic modulation, in which a binary message
modulates a parameter of the chaotic generator. This paper presents a method of
attack based on estimating the short-time period of the ciphertext generated
from the modulated chaotic attractor. By calculating and then filtering the
short-time period of the transmitted signal it is possible to obtain the binary
information signal with great accuracy without any knowledge of the parameters
of the underlying chaotic system. This method is successfully applied to
various secure communication systems proposed in the literature based on
different chaotic attractors. | nlin_CD |
Variational principles in the analysis of traffic flows. (Why it is
worth to go against the flow.): By means of a novel variational approach and using dual maps techniques and
general ideas of dynamical system theory we derive exact results about several
models of transport flows, for which we also obtain a complete description of
their limit (in time) behavior in the space of configurations. Using these
results we study the motion of a speedy passive particle (tracer) moving
along/against the flow of slow particles and demonstrate that the latter case
might be more efficient. | nlin_CD |
Rough basin boundaries in high dimension: Can we classify them
experimentally?: We show that a known condition for having rough basin boundaries in bistable
2D maps holds for high-dimensional bistable systems that possess a unique
nonattracting chaotic set embedded in their basin boundaries. The condition for
roughness is that the cross-boundary Lyapunov exponent $\lambda_x$ {\bfac on
the nonattracting set} is not the maximal one. Furthermore, we provide a
formula for the generally noninteger co-dimension of the rough basin boundary,
which can be viewed as a generalization of the Kantz-Grassberger formula. This
co-dimension that can be at most unity can be thought of as a partial
co-dimension, and, so, it can be matched with a Lyapunov exponent. We show
{\bfac in 2D noninvertible- and 3D invertible minimal models,} that, formally,
it cannot be matched with $\lambda_x$. Rather, the partial dimension
$D_0^{(x)}$ that $\lambda_x$ is associated with in the case of rough boundaries
is trivially unity. Further results hint that the latter holds also in higher
dimensions. This is a peculiar feature of rough fractals. Yet, $D_0^{(x)}$
cannot be measured via the uncertainty exponent along a line that traverses the
boundary. Indeed, one cannot determine whether the boundary is a rough or a
filamentary fractal by measuring fractal dimensions. Instead, one needs to
measure both the maximal and cross-boundary Lyapunov exponents numerically or
experimentally. | nlin_CD |
Learning to imitate stochastic time series in a compositional way by
chaos: This study shows that a mixture of RNN experts model can acquire the ability
to generate sequences combining multiple primitive patterns by means of
self-organizing chaos. By training of the model, each expert learns a primitive
sequence pattern, and a gating network learns to imitate stochastic switching
of the multiple primitives via a chaotic dynamics, utilizing a sensitive
dependence on initial conditions. As a demonstration, we present a numerical
simulation in which the model learns Markov chain switching among some
Lissajous curves by a chaotic dynamics. Our analysis shows that by using a
sufficient amount of training data, balanced with the network memory capacity,
it is possible to satisfy the conditions for embedding the target stochastic
sequences into a chaotic dynamical system. It is also shown that reconstruction
of a stochastic time series by a chaotic model can be stabilized by adding a
negligible amount of noise to the dynamics of the model. | nlin_CD |
Design of time delayed chaotic circuit with threshold controller: A novel time delayed chaotic oscillator exhibiting mono- and double scroll
complex chaotic attractors is designed. This circuit consists of only a few
operational amplifiers and diodes and employs a threshold controller for
flexibility. It efficiently implements a piecewise linear function. The control
of piecewise linear function facilitates controlling the shape of the
attractors. This is demonstrated by constructing the phase portraits of the
attractors through numerical simulations and hardware experiments. Based on
these studies, we find that this circuit can produce multi-scroll chaotic
attractors by just introducing more number of threshold values. | nlin_CD |
Fronts in passive scalar turbulence: The evolution of scalar fields transported by turbulent flow is characterized
by the presence of fronts, which rule the small-scale statistics of scalar
fluctuations. With the aid of numerical simulations, it is shown that: isotropy
is not recovered, in the classical sense, at small scales; scaling exponents
are universal with respect to the scalar injection mechanisms; high-order
exponents saturate to a constant value; non-mature fronts dominate the
statistics of intense fluctuations. Results on the statistics inside the
plateaux, where fluctuations are weak, are also presented. Finally, we analyze
the statistics of scalar dissipation and scalar fluxes. | nlin_CD |
Zero tension Kardar-Parisi-Zhang equation in (d+1)- Dimensions: The joint probability distribution function (PDF) of the height and its
gradients is derived for a zero tension $d+1$-dimensional Kardar-Parisi-Zhang
(KPZ) equation. It is proved that the height`s PDF of zero tension KPZ equation
shows lack of positivity after a finite time $t_{c}$. The properties of zero
tension KPZ equation and its differences with the case that it possess an
infinitesimal surface tension is discussed. Also potential relation between the
time scale $t_{c}$ and the singularity time scale $t_{c, \nu \to 0}$ of the KPZ
equation with an infinitesimal surface tension is investigated. | nlin_CD |
Nests and Chains of Hofstadter Butterflies: The \lq Hofstadter butterfly', a plot of the spectrum of an electron in a
two-dimensional periodic potential with a uniform magnetic field, contains
subsets which resemble small, distorted images of the entire plot. We show how
the sizes of these sub-images are determined, and calculate scaling factors
describing their self-similar nesting, revealing an un-expected simplicity in
the fractal structure of the spectrum. We also characterise semi-infinite
chains of sub-images, showing one end of the chain is a result of gap closure,
and the other end is at an accumulation point. | nlin_CD |
Generalized synchronization in mutually coupled oscillators and complex
networks: We introduce a novel concept of generalized synchronization, able to
encompass the setting of collective synchronized behavior for mutually coupled
systems and networking systems featuring complex topologies in their
connections. The onset of the synchronous regime is confirmed by the dependence
of the system's Lyapunov exponents on the coupling parameter. The presence of a
generalized synchronization regime is verified by means of the nearest neighbor
method. | nlin_CD |
Transition from amplitude to oscillation death under mean-field
diffusive coupling: We study the transition from amplitude death (AD) to oscillation death (OD)
state in limit-cycle oscillators coupled through mean-field diffusion. We show
that this coupling scheme can induce an important transition from AD to OD even
in {\it identical} limit cycle oscillators. We identify a parameter region
where OD and a novel {\it nontrivial} AD (NT-AD) state coexist. This NT-AD
state is unique in comparison with AD owing to the fact that it is created by a
subcritical pitchfork bifurcation, and parameter mismatch does not support but
destroy this state. We extend our study to a network of mean-field coupled
oscillators to show that the transition scenario preserves and the oscillators
form a two cluster state. | nlin_CD |
Evanescent wave approach to diffractive phenomena in convex billiards
with corners: What we are going to call in this paper "diffractive phenomena" in billiards
is far from being deeply understood. These are sorts of singularities that, for
example, some kind of corners introduce in the energy eigenfunctions. In this
paper we use the well-known scaling quantization procedure to study them. We
show how the scaling method can be applied to convex billiards with corners,
taking into account the strong diffraction at them and the techniques needed to
solve their Helmholtz equation. As an example we study a classically
pseudointegrable billiard, the truncated triangle. Then we focus our attention
on the spectral behavior. A numerical study of the statistical properties of
high-lying energy levels is carried out. It is found that all computed
statistical quantities are roughly described by the so-called semi-Poisson
statistics, but it is not clear whether the semi-Poisson statistics is the
correct one in the semiclassical limit. | nlin_CD |
Symbolic dynamics techniques for complex systems: Application to share
price dynamics: The symbolic dynamics technique is well-known for low-dimensional dynamical
systems and chaotic maps, and lies at the roots of the thermodynamic formalism
of dynamical systems. Here we show that this technique can also be successfully
applied to time series generated by complex systems of much higher
dimensionality. Our main example is the investigation of share price returns in
a coarse-grained way. A nontrivial spectrum of Renyi entropies is found. We
study how the spectrum depends on the time scale of returns, the sector of
stocks considered, as well as the number of symbols used for the symbolic
description. Overall our analysis confirms that in the symbol space transition
probabilities of observed share price returns depend on the entire history of
previous symbols, thus emphasizing the need for a modelling based on
non-Markovian stochastic processes. Our method allows for quantitative
comparisons of entirely different complex systems, for example the statistics
of symbol sequences generated by share price returns using 4 symbols can be
compared with that of genomic sequences. | nlin_CD |
One-Particle and Few-Particle Billiards: We study the dynamics of one-particle and few-particle billiard systems in
containers of various shapes. In few-particle systems, the particles collide
elastically both against the boundary and against each other. In the
one-particle case, we investigate the formation and destruction of resonance
islands in (generalized) mushroom billiards, which are a recently discovered
class of Hamiltonian systems with mixed regular-chaotic dynamics. In the
few-particle case, we compare the dynamics in container geometries whose
counterpart one-particle billiards are integrable, chaotic, and mixed. One of
our findings is that two-, three-, and four-particle billiards confined to
containers with integrable one-particle counterparts inherit some integrals of
motion and exhibit a regular partition of phase space into ergodic components
of positive measure. Therefore, the shape of a container matters not only for
noninteracting particles but also for interacting particles. | nlin_CD |
Ehrenfest times for classically chaotic systems: We describe the quantum mechanical spreading of a Gaussian wave packet by
means of the semiclassical WKB approximation of Berry and Balazs. We find that
the time scale $\tau$ on which this approximation breaks down in a chaotic
system is larger than the Ehrenfest times considered previously. In one
dimension $\tau=\fr{7}{6}\lambda^{-1}\ln(A/\hbar)$, with $\lambda$ the Lyapunov
exponent and $A$ a typical classical action. | nlin_CD |
Mapping Model of Chaotic Phase Synchronization: A coupled map model for the chaotic phase synchronization and its
desynchronization phenomenon is proposed. The model is constructed by
integrating the coupled kicked oscillator system, kicking strength depending on
the complex state variables. It is shown that the proposed model clearly
exhibits the chaotic phase synchronization phenomenon. Furthermore, we
numerically prove that in the region where the phase synchronization is weakly
broken, the anomalous scaling of the phase difference rotation number is
observed. This proves that the present model belongs to the same universality
class found by Pikovsky et al.. Furthermore, the phase diffusion coefficient in
the de-synchronization state is analyzed. | nlin_CD |
Boundary-induced instabilities in coupled oscillators: A novel class of nonequilibrium phase-transitions at zero temperature is
found in chains of nonlinear oscillators.For two paradigmatic systems, the
Hamiltonian XY model and the discrete nonlinear Schr\"odinger equation, we find
that the application of boundary forces induces two synchronized phases,
separated by a non-trivial interfacial region where the kinetic temperature is
finite. Dynamics in such supercritical state displays anomalous chaotic
properties whereby some observables are non-extensive and transport is
superdiffusive. At finite temperatures, the transition is smoothed, but the
temperature profile is still non-monotonous. | nlin_CD |
Python for Education: Computational Methods for Nonlinear Systems: We describe a novel, interdisciplinary, computational methods course that
uses Python and associated numerical and visualization libraries to enable
students to implement simulations for a number of different course modules.
Problems in complex networks, biomechanics, pattern formation, and gene
regulation are highlighted to illustrate the breadth and flexibility of
Python-powered computational environments. | nlin_CD |
The smallest chimera states: We demonstrate that chimera behavior can be observed in small networks
consisting of three identical oscillators, with mutual all-to-all coupling.
Three different types of chimeras, characterized by the coexistence of two
coherent oscillators and one incoherent oscillator (i.e. rotating with another
frequency) have been identified, where the oscillators show periodic (two
types) and chaotic (one type) behaviors. Typical bifurcations at the
transitions from full synchronization to chimera states and between different
types of chimeras have been described. Parameter regions for the chimera states
are obtained in the form of Arnold tongues, issued from a singular parameter
point. Our analysis suggests that chimera states can be observed in small
networks, relevant to various real-world systems. | nlin_CD |
On ergodicity for multi-dimensional harmonic oscillator systems with
Nose-Hoover type thermostat: A simple proof and detailed analysis on the non-ergodicity for
multidimensional harmonic oscillator systems with Nose-Hoover type thermostat
are given. The origin of the nonergodicity is symmetries in the
multidimensional target physical system, and is differ from that in the
Nose-Hoover thermostat with the 1-dimensional harmonic oscillator. A new simple
deterministic method to recover the ergodicity is also presented. An individual
thermostat variable is attached to each degree of freedom, and all these
variables act on a friction coefficient for each degree of freedom. This action
is linear and controlled by a Nos\'e mass matrix Q, which is a matrix analogue
of the scalar Nos\'e's mass. Matrix Q can break the symmetry and contribute to
attain the ergodicity. | nlin_CD |
Goos-Haenchen shift and localization of optical modes in deformed
microcavities: Recently, an interesting phenomenon of spatial localization of optical modes
along periodic ray trajectories near avoided resonance crossings has been
observed [J. Wiersig, Phys. Rev. Lett. 97, 253901 (2006)]. For the case of a
microdisk cavity with elliptical cross section we use the Husimi function to
analyse this localization in phase space. Moreover, we present a semiclassical
explanation of this phenomenon in terms of the Goos-Haenchen shift which works
very well even deep in the wave regime. This semiclassical correction to the
ray dynamics modifies the phase space structure such that modes can localize
either on stable islands or along unstable periodic ray trajectories. | nlin_CD |
Dynamics of traffic jams: order and chaos: By means of a novel variational approach we study ergodic properties of a
model of a multi lane traffic flow, considered as a (deterministic) wandering
of interacting particles on an infinite lattice. For a class of initial
configurations of particles (roughly speaking satisfying the Law of Large
Numbers) the complete description of their limit (in time) behavior is
obtained, as well as estimates of the transient period. In this period the main
object of interest is the dynamics of `traffic jams', which is rigorously
defined and studied. It is shown that the dynamical system under consideration
is chaotic in a sense that its topological entropy (calculated explicitly) is
positive. Statistical quantities describing limit configurations are obtained
as well. | nlin_CD |
Finite-size effects on open chaotic advection: We study the effects of finite-sizeness on small, neutrally buoyant,
spherical particles advected by open chaotic flows. We show that, when
projected onto configuration space, the advected finite-size particles disperse
about the unstable manifold of the chaotic saddle that governs the passive
advection. Using a discrete-time system for the dynamics, we obtain an
expression predicting the dispersion of the finite-size particles in terms of
their Stokes parameter at the onset of the finite-sizeness induced dispersion.
We test our theory in a system derived from a flow and find remarkable
agreement between our expression and the numerically measured dispersion. | nlin_CD |
Dynamics of the Shapovalov mid-size firm model: One of the main tasks in the study of financial and economic processes is
forecasting and analysis of the dynamics of these processes. Within this task
lie important research questions including how to determine the qualitative
properties of the dynamics and how best to estimate quantitative indicators.
These questions can be studied both empirically and theoretically. In the
empirical approach, one considers the real data represented by time series,
identifies patterns of their dynamics, and then forecasts short- and long-term
behavior of the process. The second approach is based on postulating the laws
of dynamics for the process, deriving mathematical dynamic models based on
these laws, and conducting subsequent analytical investigation of the dynamics
generated by the models.
To implement these approaches, both numerical and analytical methods can be
used. It should be noted that while numerical methods make it possible to study
complex models, the possibility of obtaining reliable results using them is
significantly limited due to calculations being performed only over finite-time
intervals, numerical errors, and the unbounded space of initial data sets. In
turn, analytical methods allow researchers to overcome these problems and to
obtain exact qualitative and quantitative characteristics of the process
dynamics. However, their effective applications are often limited to
low-dimensional models. In this paper, we develop analytical methods for the
study of deterministic dynamic systems. These methods make it possible not only
to obtain analytical stability criteria and to estimate limiting behavior, but
also to overcome the difficulties related to implementing reliable numerical
analysis of quantitative indicators. We demonstrate the effectiveness of the
proposed methods using the mid-size firm model suggested recently by V.I.
Shapovalov. | nlin_CD |
Dynamical epidemic suppression using stochastic prediction and control: We consider the effects of noise on a model of epidemic outbreaks, where the
outbreaks appear. randomly. Using a constructive transition approach that
predicts large outbreaks, prior to their occurrence, we derive an adaptive
control. scheme that prevents large outbreaks from occurring. The theory
inapplicable to a wide range of stochastic processes with underlying
deterministic structure. | nlin_CD |
Finite-time synchronization of non-autonomous chaotic systems with
unknown parameters: Adaptive control technique is adopted to synchronize two identical
non-autonomous systems with unknown parameters in finite time. A virtual
unknown parameter is introduced in order to avoid the unknown parameters from
appearing in the controllers and parameters update laws. The Duffing equation
and a gyrostat system are chosen as the numerical examples to show the validity
of the present method. | nlin_CD |
Amplitude distribution of eigenfunctions in mixed systems: We study the amplitude distribution of irregular eigenfunctions in systems
with mixed classical phase space. For an appropriately restricted random wave
model a theoretical prediction for the amplitude distribution is derived and
good agreement with numerical computations for the family of limacon billiards
is found. The natural extension of our result to more general systems, e.g.
with a potential, is also discussed. | nlin_CD |
Combined effects of compressibility and helicity on the scaling regimes
of a passive scalar advected by turbulent velocity field with finite
correlation time: The influence of compressibility and helicity on the stability of the scaling
regimes of a passive scalar advected by a Gaussian velocity field with finite
correlation time is investigated by the field theoretic renormalization group
within two-loop approximation. The influence of helicity and compressibility on
the scaling regimes is discussed as a function of the exponents $\epsilon$ and
$\eta$, where $\epsilon$ characterizes the energy spectrum of the velocity
field in the inertial range $E\propto k^{1-2\epsilon}$, and $\eta$ is related
to the correlation time at the wave number $k$ which is scaled as
$k^{-2+\eta}$. The restrictions given by nonzero compressibility and helicity
on the regions with stable infrared fixed points which correspond to the stable
infrared scaling regimes are discussed. A special attention is paid to the case
of so-called frozen velocity field when the velocity correlator is time
independent. In this case, explicit inequalities which must be fulfilled in the
plane $\epsilon-\eta$ are determined within two-loop approximation. | nlin_CD |
Transition from homogeneous to inhomogeneous limit cycles: Effect of
local filtering in coupled oscillators: We report an interesting symmetry-breaking transition in coupled identical
oscillators, namely the continuous transition from homogeneous to inhomogeneous
limit cycle oscillations. The observed transition is the oscillatory analog of
the Turing-type symmetry-breaking transition from amplitude death (i.e., stable
homogeneous steady state) to oscillation death (i.e., stable inhomogeneous
steady state). This novel transition occurs in the parametric zone of
occurrence of rhythmogenesis and oscillation death as a consequence of the
presence of local filtering in the coupling path. We consider paradigmatic
oscillators, such as Stuart-Landau and van der Pol oscillators under mean-field
coupling with low-pass or all-pass filtered self-feedback and through a
rigorous bifurcation analysis we explore the genesis of this transition.
Further, we experimentally demonstrate the observed transition, which
establishes its robustness in the presence of parameter fluctuations and noise. | nlin_CD |
Community structure in real-world networks from a non-parametrical
synchronization-based dynamical approach: This work analyzes the problem of community structure in real-world networks
based on the synchronization of nonidentical coupled chaotic R\"{o}ssler
oscillators each one characterized by a defined natural frequency, and coupled
according to a predefined network topology. The interaction scheme contemplates
an uniformly increasing coupling force to simulate a society in which the
association between the agents grows in time. To enhance the stability of the
correlated states that could emerge from the synchronization process, we
propose a parameterless mechanism that adapts the characteristic frequencies of
coupled oscillators according to a dynamic connectivity matrix deduced from
correlated data. We show that the characteristic frequency vector that results
from the adaptation mechanism reveals the underlying community structure
present in the network. | nlin_CD |
Quantitative predictions with detuned normal forms: The phase-space structure of two families of galactic potentials is
approximated with a resonant detuned normal form. The normal form series is
obtained by a Lie transform of the series expansion around the minimum of the
original Hamiltonian. Attention is focused on the quantitative predictive
ability of the normal form. We find analytical expressions for bifurcations of
periodic orbits and compare them with other analytical approaches and with
numerical results. The predictions are quite reliable even outside the
convergence radius of the perturbation and we analyze this result using
resummation techniques of asymptotic series. | nlin_CD |
Multistability in Piecewise Linear Systems by Means of the Eigenspectra
Variation and the Round Function: A multistable system generated by a Piecewise Linear (PWL) system based on
the jerky equation is presented. The systems behaviour is characterised by
means of the Nearest Integer or round(x) function to control the switching
events and to locate the corresponding equilibria among each of the commutation
surfaces. These surfaces are generated by means of the switching function
dividing the space in regions equally distributed along one axis. The
trajectory of this type of system is governed by the eigenspectra of the
coefficient matrix which can be adjusted by means of a bifurcation parameter.
The behaviour of the system can change from multi-scroll attractors into a
mono-stable state to the coexistence of several single-scroll attractors into a
multi-stable state. Numerical results of the dynamics and bifurcation analyses
of their parameters are displayed to depict the multi-stable states. | nlin_CD |
Lyapunov exponent for inertial particles in the 2D Kraichnan model as a
problem of Anderson localization with complex valued potential: We exploit the analogy between dynamics of inertial particle pair separation
in a random-in-time flow and the Anderson model of a quantum particle on the
line in a spatially random real-valued potential. Thereby we get an exact
formula for the Lyapunov exponent of pair separation in a special case, and we
are able to generalize the class of solvable models slightly, for potentials
that are real up to a global complex multiplier. A further important result for
inertial particle behavior, supported by analytical computations in some cases
and by numerics more generally, is that of the decay of the Lyapunov exponent
with large Stokes number (quotient of particle relaxation and flow turn-over
time-scales) as Stokes number to the power -2/3. | nlin_CD |
Langevin approach to synchronization of hyperchaotic time-delay dynamics: In this paper, we characterize the synchronization phenomenon of hyperchaotic
scalar non-linear delay dynamics in a fully-developed chaos regime. Our results
rely on the observation that, in that regime, the stationary statistical
properties of a class of hyperchaotic attractors can be reproduced with a
linear Langevin equation, defined by replacing the non-linear delay force by a
delta-correlated noise. Therefore, the synchronization phenomenon can be
analytically characterized by a set of coupled Langevin equations. We apply
this formalism to study anticipated synchronization dynamics subject to
external noise fluctuations as well as for characterizing the effects of
parameter mismatch in a hyperchaotic communication scheme. The same procedure
is applied to second order differential delay equations associated to
synchronization in electro-optical devices. In all cases, the departure with
respect to perfect synchronization is measured through a similarity function.
Numerical simulations in discrete maps associated to the hyperchaotic dynamics
support the formalism. | nlin_CD |
Chaos in Nonlinear Random Walks with Non-Monotonic Transition
Probabilities: Random walks serve as important tools for studying complex network
structures, yet their dynamics in cases where transition probabilities are not
static remain under explored and poorly understood. Here we study nonlinear
random walks that occur when transition probabilities depend on the state of
the system. We show that when these transition probabilities are non-monotonic,
i.e., are not uniformly biased towards the most densely or sparsely populated
nodes, but rather direct random walkers with more nuance, chaotic dynamics
emerge. Using multiple transition probability functions and a range of networks
with different connectivity properties, we demonstrate that this phenomenon is
generic. Thus, when such non-monotonic properties are key ingredients in
nonlinear transport applications complicated and unpredictable behaviors may
result. | nlin_CD |
Bifurcation analysis and chaos control of periodically driven discrete
fractional order memristive Duffing Oscillator: Discrete fractional order chaotic systems extends the memory capability to
capture the discrete nature of physical systems. In this research, the
memristive discrete fractional order chaotic system is introduced. The dynamics
of the system was studied using bifurcation diagrams and phase space
construction. The system was found chaotic with fractional order
$0.465<n<0.562$. The dynamics of the system under different values makes it
useful as a switch. Controllers were developed for the tracking control of the
two systems to different trajectories. The effectiveness of the designed
controllers were confirmed using simulations | nlin_CD |
Aspects of the Scattering and Impedance Properties of Chaotic Microwave
Cavities: We consider the statistics of the impedance of a chaotic microwave cavity
coupled to a single port. We remove the non-universal effects of the coupling
from the experimental data using the radiation impedance obtained directly from
the experiments. We thus obtain the normalized impedance whose Probability
Density Function (PDF) is predicted to be universal in that it depends only on
the loss (quality factor) of the cavity. We find that impedance fluctuations
decrease with increasing loss. The results apply to scattering measurements on
any wave chaotic system. | nlin_CD |
Experimental control of chaos by variable and distributed delay feedback: We report on a significant improvement of the classical time-delayed feedback
control method for stabilization of unstable periodic orbits or steady states.
In an electronic circuit experiment we were able to realize time-varying and
distributed delays in the control force leading to successful control for large
parameter sets including large time delays. The presented technique makes
advanced use of the natural torsion of the orbits, which is also necessary for
the original control method to work. | nlin_CD |
Diffusion For Ensembles of Standard Maps: Two types of random evolution processes are studied for ensembles of the
standard map with driving parameter $K$ that determines its degree of
stochasticity. For one type of processes the parameter $K$ is chosen at random
from a Gaussian distribution and is then kept fixed, while for the other type
it varies from step to step. In addition, noise that can be arbitrarily weak is
added. The ensemble average and the average over noise of the diffusion
coefficient is calculated for both types of processes. These two types of
processes are relevant for two types of experimental situations as explained in
the paper. Both types of processes destroy fine details of the dynamics, and
the second process is found to be more effective in destroying the fine
details. We hope that this work is a step in the efforts for developing a
statistical theory for systems with mixed phase space (regular in some parts
and chaotic in other parts). | nlin_CD |
Quantum Dynamical Tunneling Breaks Classical Conserved Quantities: We discover that quantum dynamical tunneling, occurring between phase space
regions in a classically forbidden way, can break conserved quantities in
pseudointegrable systems. We rigorously prove that a conserved quantity in a
class of typical pseudointegrable systems can be broken quantum mechanically.
We then numerically compute the uncertainties of this broken conserved
quantity, which remain non-zero for up to $10^5$ eigenstates and exhibit
universal distributions similar to energy level statistics. Furthermore, all
the eigenstates with large uncertainties show the superpositions of regular
orbits with different values of the conserved quantity, showing definitive
manifestation of dynamical tunneling. A random matrix model is constructed to
successfully reproduce the level statistics in pseudointegrable systems. | nlin_CD |
Sensitivity to perturbations in a quantum chaotic billiard: The Loschmidt echo (LE) measures the ability of a system to return to the
initial state after a forward quantum evolution followed by a backward
perturbed one. It has been conjectured that the echo of a classically chaotic
system decays exponentially, with a decay rate given by the minimum between the
width $\Gamma$ of the local density of states and the Lyapunov exponent. As the
perturbation strength is increased one obtains a cross-over between both
regimes. These predictions are based on situations where the Fermi Golden Rule
(FGR) is valid. By considering a paradigmatic fully chaotic system, the
Bunimovich stadium billiard, with a perturbation in a regime for which the FGR
manifestly does not work, we find a cross over from $\Gamma$ to Lyapunov decay.
We find that, challenging the analytic interpretation, these conjetures are
valid even beyond the expected range. | nlin_CD |
Chaotic Diffusion in Delay Systems: Giant Enhancement by Time Lag
Modulation: We consider a typical class of systems with delayed nonlinearity, which we
show to exhibit chaotic diffusion. It is demonstrated that a periodic
modulation of the time-lag can lead to an enhancement of the diffusion constant
by several orders of magnitude. This effect is the largest if the circle map
defined by the modulation shows mode locking and more specifically, fulfills
the conditions for laminar chaos. Thus we establish for the first time a
connection between Arnold tongue structures in parameter space and diffusive
properties of a delay system. Counterintuitively, the enhancement of diffusion
is accompanied by a strong reduction of the effective dimensionality of the
system. | nlin_CD |
Coherent Response in a Chaotic Neural Network: We set up a signal-driven scheme of the chaotic neural network with the
coupling constants corresponding to certain information, and investigate the
stochastic resonance-like effects under its deterministic dynamics, comparing
with the conventional case of Hopfield network with stochastic noise. It is
shown that the chaotic neural network can enhance weak subthreshold signals and
have higher coherence abilities between stimulus and response than those
attained by the conventional stochastic model. | nlin_CD |
Classical projected phase space density of billiards and its relation to
the quantum Neumann spectrum: A comparison of classical and quantum evolution usually involves a
quasi-probability distribution as a quantum analogue of the classical phase
space distribution. In an alternate approach that we adopt here, the classical
density is projected on to the configuration space. We show that for billiards,
the eigenfunctions of the coarse-grained projected classical evolution operator
are identical to a first approximation to the quantum Neumann eigenfunctions.
However, even though there exists a correspondence between the respective
eigenvalues, their time evolutions differ. This is demonstrated numerically for
the stadium and lemon shaped billiards. | nlin_CD |
Critical States and Fractal Attractors in Fractal Tongues: Localization
in the Harper map: Localized states of Harper's equation correspond to strange nonchaotic
attractors (SNAs) in the related Harper mapping. In parameter space, these
fractal attractors with nonpositive Lyapunov exponents occur in fractally
organized tongue--like regions which emanate from the Cantor set of eigenvalues
on the critical line $\epsilon = 1$. A topological invariant characterizes
wavefunctions corresponding to energies in the gaps in the spectrum. This
permits a unique integer labeling of the gaps and also determines their scaling
properties as a function of potential strength. | nlin_CD |
Soft billiards with corners: We develop a framework for dealing with smooth approximations to billiards
with corners in the two-dimensional setting. Let a polygonal trajectory in a
billiard start and end up at the same billiard's corner point. We prove that
smooth Hamiltonian flows which limit to this billiard have a nearby periodic
orbit if and only if the polygon angles at the corner are ''acceptable''. The
criterion for a corner polygon to be acceptable depends on the smooth potential
behavior at the corners, which is expressed in terms of a {scattering
function}. We define such an asymptotic scattering function and prove the
existence of it, explain how it can be calculated and predict some of its
properties. In particular, we show that it is non-monotone for some potentials
in some phase space regions. We prove that when the smooth system has a
limiting periodic orbit it is hyperbolic provided the scattering function is
not extremal there. We then prove that if the scattering function is extremal,
the smooth system has elliptic periodic orbits limiting to the corner polygon,
and, furthermore, that the return map near these periodic orbits is conjugate
to a small perturbation of the Henon map and therefore has elliptic islands. We
find from the scaling that the island size is typically algebraic in the
smoothing parameter and exponentially small in the number of reflections of the
polygon orbit. | nlin_CD |
To what extent can dynamical models describe statistical features of
turbulent flows?: Statistical features of "bursty" behaviour in charged and neutral fluid
turbulence, are compared to statistics of intermittent events in a GOY shell
model, and avalanches in different models of Self Organized Criticality (SOC).
It is found that inter-burst times show a power law distribution for turbulent
samples and for the shell model, a property which is shared only in a
particular case of the running sandpile model. The breakdown of self-similarity
generated by isolated events observed in the turbulent samples, is well
reproduced by the shell model, while it is absent in all SOC models considered.
On this base, we conclude that SOC models are not adequate to mimic fluid
turbulence, while the GOY shell model constitutes a better candidate to
describe the gross features of turbulence. | nlin_CD |
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