text
stringlengths 73
2.82k
| category
stringclasses 21
values |
|---|---|
Visualization of Four Limit Cycles in Near-Integrable Quadratic
Polynomial Systems: It has been known for almost $40$ years that general planar quadratic
polynomial systems can have four limit cycles. Recently, four limit cycles were
also found in near-integrable quadratic polynomial systems. To help more people
to understand limit cycles theory, the visualization of such four numerically
simulated limit cycles in quadratic systems has attracted researchers'
attention. However, for near integral systems, such visualization becomes much
more difficult due to limitation on choosing parameter values. In this paper,
we start from the simulation of the well-known quadratic systems constructed
around the end of 1979, then reconsider the simulation of a recently published
quadratic system which exhibits four big size limit cycles, and finally provide
a concrete near-integral quadratic polynomial system to show four normal size
limit cycles. | nlin_CD |
Ruelle-Perron-Frobenius spectrum for Anosov maps: We extend a number of results from one dimensional dynamics based on spectral
properties of the Ruelle-Perron-Frobenius transfer operator to Anosov
diffeomorphisms on compact manifolds. This allows to develop a direct operator
approach to study ergodic properties of these maps. In particular, we show that
it is possible to define Banach spaces on which the transfer operator is
quasicompact. (Information on the existence of an SRB measure, its smoothness
properties and statistical properties readily follow from such a result.) In
dimension $d=2$ we show that the transfer operator associated to smooth random
perturbations of the map is close, in a proper sense, to the unperturbed
transfer operator. This allows to obtain easily very strong spectral stability
results, which in turn imply spectral stability results for smooth
deterministic perturbations as well. Finally, we are able to implement an Ulam
type finite rank approximation scheme thus reducing the study of the spectral
properties of the transfer operator to a finite dimensional problem. | nlin_CD |
Exact Analysis of the Adiabatic Invariants in Time-Dependent Harmonic
Oscillator: The theory of adiabatic invariants has a long history and important
applications in physics but is rarely rigorous. Here we treat exactly the
general time-dependent 1-D harmonic oscillator, $\ddot{q} + \omega^2(t) q=0$
which cannot be solved in general. We follow the time-evolution of an initial
ensemble of phase points with sharply defined energy $E_0$ and calculate
rigorously the distribution of energy $E_1$ after time $T$, and all its
moments, especially its average value $\bar{E_1}$ and variance $\mu^2$. Using
our exact WKB-theory to all orders we get the exact result for the leading
asymptotic behaviour of $\mu^2$. | nlin_CD |
Determining functionals for random partial differential equations: Determining functionals are tools to describe the finite dimensional
long-term dynamics of infinite dimensional dynamical systems. There also exist
several applications to infinite dimensional {\em random} dynamical systems. In
these applications the convergence condition of the trajectories of an infinite
dimensional random dynamical system with respect to a finite set of linear
functionals is assumed to be either in mean or exponential with respect to the
convergence almost surely. In contrast to these ideas we introduce a
convergence concept which is based on the convergence in probability. By this
ansatz we get rid of the assumption of exponential convergence. In addition,
setting the random terms to zero we obtain usual deterministic results. We
apply our results to the 2D Navier - Stokes equations forced by a white noise. | nlin_CD |
Infinite Products of Random Isotropically Distributed Matrices: Statistical properties of infinite products of random isotropically
distributed matrices are investigated. Both for continuous processes with
finite correlation time and discrete sequences of independent matrices, a
formalism that allows to calculate easily the Lyapunov spectrum and generalized
Lyapunov exponents is developed. This problem is of interest to probability
theory, statistical characteristics of matrix T-exponentials are also needed
for turbulent transport problems, dynamical chaos and other parts of
statistical physics. | nlin_CD |
Fokker - Planck equation in curvilinear coordinates. Part 2: The aim of this paper is to derive Fokker - Planck equation in curvilinear
coordinates using physical argumentation. We get the same result, as in our
previous article [1], but for broader class of arbitrary holonomic mechanical
systems. | nlin_CD |
Control of stochasticity in magnetic field lines: We present a method of control which is able to create barriers to magnetic
field line diffusion by a small modification of the magnetic perturbation. This
method of control is based on a localized control of chaos in Hamiltonian
systems. The aim is to modify the perturbation locally by a small control term
which creates invariant tori acting as barriers to diffusion for Hamiltonian
systems with two degrees of freedom. The location of the invariant torus is
enforced in the vicinity of the chosen target. Given the importance of
confinement in magnetic fusion devices, the method is applied to two examples
with a loss of magnetic confinement. In the case of locked tearing modes, an
invariant torus can be restored that aims at showing the current quench and
therefore the generation of runaway electrons. In the second case, the method
is applied to the control of stochastic boundaries allowing one to define a
transport barrier within the stochastic boundary and therefore to monitor the
volume of closed field lines. | nlin_CD |
Conditional entropy of ordinal patterns: In this paper we investigate a quantity called conditional entropy of ordinal
patterns, akin to the permutation entropy. The conditional entropy of ordinal
patterns describes the average diversity of the ordinal patterns succeeding a
given ordinal pattern. We observe that this quantity provides a good estimation
of the Kolmogorov-Sinai entropy in many cases. In particular, the conditional
entropy of ordinal patterns of a finite order coincides with the
Kolmogorov-Sinai entropy for periodic dynamics and for Markov shifts over a
binary alphabet. Finally, the conditional entropy of ordinal patterns is
computationally simple and thus can be well applied to real-world data. | nlin_CD |
Phase reduction of a limit cycle oscillator perturbed by a strong
amplitude-modulated high-frequency force: The phase reduction method for a limit cycle oscillator subjected to a strong
amplitude-modulated high-frequency force is developed. An equation for the
phase dynamics is derived by introducing a new, effective phase response curve.
We show that if the effective phase response curve is everywhere positive
(negative), then an entrainment of the oscillator to an envelope frequency is
possible only when this frequency is higher (lower) than the natural frequency
of the oscillator. Also, by using the Pontryagin maximum principle, we have
derived an optimal waveform of the perturbation that ensures an entrainment of
the oscillator with minimal power. The theoretical results are demonstrated
with the Stuart-Landau oscillator and model neurons. | nlin_CD |
Bifurcation Diagrams and Generalized Bifurcation Diagrams for a
rotational model of an oblate satellite: This paper presents bifurcation and generalized bifurcation diagrams for a
rotational model of an oblate satellite. Special attention is paid to parameter
values describing one of Saturn's moons, Hyperion. For various oblateness the
largest Lyapunov Characteristic Exponent (LCE) is plotted. The largest LCE in
the initial condition as well as in the mixed parameter-initial condition space
exhibits a fractal structure, for which the fractal dimension was calculated.
It results from the bifurcation diagrams of which most of the parameter values
for preselected initial conditions lead to chaotic rotation. The First
Recurrence Time (FRT) diagram provides an explanation of the birth of chaos and
the existence of quasi-periodic windows occuring in the bifurcation diagrams. | nlin_CD |
Analyzing intramolecular dynamics by Fast Lyapunov Indicators: We report an analysis of intramolecular dynamics of the highly excited planar
carbonyl sulfide (OCS) below and at the dissociation threshold by the Fast
Lyapunov Indicator (FLI) method. By mapping out the variety of dynamical
regimes in the phase space of this molecule, we obtain the degree of regularity
of the system versus its energy. We combine this stability analysis with a
periodic orbit search, which yields a family of elliptic periodic orbits in the
regular part of phase space an a family of in-phase collinear hyperbolic orbits
associated with the chaotic regime. | nlin_CD |
Statistics and geometry of passive scalars in turbulence: We present direct numerical simulations (DNS) of the mixing of the passive
scalar at modest Reynolds numbers (10 =< R_\lambda =< 42) and Schmidt numbers
larger than unity (2 =< Sc =< 32). The simulations resolve below the Batchelor
scale up to a factor of four. The advecting turbulence is homogeneous and
isotropic, and maintained stationary by stochastic forcing at low wavenumbers.
The passive scalar is rendered stationary by a mean scalar gradient in one
direction. The relation between geometrical and statistical properties of
scalar field and its gradients is examined. The Reynolds numbers and Schmidt
numbers are not large enough for either the Kolmogorov scaling or the Batchelor
scaling to develop and, not surprisingly, we find no fractal scaling of scalar
level sets, or isosurfaces, in the intermediate viscous range. The
area-to-volume ratio of isosurfaces reflects the nearly Gaussian statistics of
the scalar fluctuations. The scalar flux across the isosurfaces, which is
determined by the conditional probability density function (PDF) of the scalar
gradient magnitude, has a stretched exponential distribution towards the tails.
The PDF of the scalar dissipation departs distinctly, for both small and large
amplitudes, from the lognormal distribution for all cases considered. The joint
statistics of the scalar and its dissipation rate, and the mean conditional
moment of the scalar dissipation, are studied as well. We examine the effects
of coarse-graining on the probability density to simulate the effects of poor
probe-resolution in measurements. | nlin_CD |
Spectral decomposition of 3D Fokker - Planck differential operator: We construct spectral decomposition of 3D Fokker - Planck differential
operator in this paper. We use the decomposition to obtain solution of Cauchy
problem - and especially the fundamental solution. Then we use the
decomposition to calculate macroscopic parameters of Fokker - Planck flow. | nlin_CD |
A New Family of Generalized 3D Cat Maps: Since the 1990s chaotic cat maps are widely used in data encryption, for
their very complicated dynamics within a simple model and desired
characteristics related to requirements of cryptography. The number of cat map
parameters and the map period length after discretization are two major
concerns in many applications for security reasons. In this paper, we propose a
new family of 36 distinctive 3D cat maps with different spatial configurations
taking existing 3D cat maps [1]-[4] as special cases. Our analysis and
comparisons show that this new 3D cat maps family has more independent map
parameters and much longer averaged period lengths than existing 3D cat maps.
The presented cat map family can be extended to higher dimensional cases. | nlin_CD |
Lagrangian particle paths and ortho-normal quaternion frames: Experimentalists now measure intense rotations of Lagrangian particles in
turbulent flows by tracking their trajectories and Lagrangian-average velocity
gradients at high Reynolds numbers. This paper formulates the dynamics of an
orthonormal frame attached to each Lagrangian fluid particle undergoing
three-axis rotations, by using quaternions in combination with Ertel's theorem
for frozen-in vorticity. The method is applicable to a wide range of Lagrangian
flows including the three-dimensional Euler equations and its variants such as
ideal MHD. The applicability of the quaterionic frame description to Lagrangian
averaged velocity gradient dynamics is also demonstrated. | nlin_CD |
Instantaneous frequencies in the Kuramoto model: Using the main results of the Kuramoto theory of globally coupled phase
oscillators combined with methods from probability and generalized function
theory in a geometric analysis, we extend Kuramoto's results and obtain a
mathematical description of the instantaneous frequency (phase-velocity)
distribution. Our result is validated against numerical simulations, and we
illustrate it in cases where the natural frequencies have normal and Beta
distributions. In both cases, we vary the coupling strength and compare
systematically the distribution of time-averaged frequencies (a known result of
Kuramoto theory) to that of instantaneous frequencies, focussing on their
qualitative differences near the synchronized frequency and in their tails. For
a class of natural frequency distributions with power-law tails, which includes
the Cauchy-Lorentz distribution, we analyze rare events by means of an
asymptotic formula obtained from a power series expansion of the instantaneous
frequency distribution. | nlin_CD |
Classical dynamics and particle transport in kicked billiards: We study nonlinear dynamics of the kicked particle whose motion is confined
by square billiard. The kick source is considered as localized at the center of
square with central symmetric spatial distribution. It is found that ensemble
averaged energy of the particle diffusively grows as a function of time. This
growth is much more extensive than that of kicked rotor energy. It is shown
that momentum transfer distribution in kicked billiard is considerably
different than that for kicked free particle. Time-dependence of the ensemble
averaged energy for different localizations of the kick source is also
explored. It is found that changing of localization doesn't lead to crucial
changes in the time-dependence of the energy. Also, escape and transport of
particles are studied by considering kicked open billiard with one and three
holes, respectively. It is found that for the open billiard with one hole the
number of (non-interacting) billiard particles decreases according to
exponential law. | nlin_CD |
Testing the assumptions of linear prediction analysis in normal vowels: This paper develops an improved surrogate data test to show experimental
evidence, for all the simple vowels of US English, for both male and female
speakers, that Gaussian linear prediction analysis, a ubiquitous technique in
current speech technologies, cannot be used to extract all the dynamical
structure of real speech time series. The test provides robust evidence
undermining the validity of these linear techniques, supporting the assumptions
of either dynamical nonlinearity and/or non-Gaussianity common to more recent,
complex, efforts at dynamical modelling speech time series. However, an
additional finding is that the classical assumptions cannot be ruled out
entirely, and plausible evidence is given to explain the success of the linear
Gaussian theory as a weak approximation to the true, nonlinear/non-Gaussian
dynamics. This supports the use of appropriate hybrid
linear/nonlinear/non-Gaussian modelling. With a calibrated calculation of
statistic and particular choice of experimental protocol, some of the known
systematic problems of the method of surrogate data testing are circumvented to
obtain results to support the conclusions to a high level of significance. | nlin_CD |
hbar expansions in semiclassical theories for systems with smooth
potentials and discrete symmetries: We extend a theory of first order hbar corrections to Gutzwiller's trace
formula for systems with a smooth potential to systems with discrete symmetries
and, as an example, apply the method to the two-dimensional hydrogen atom in a
uniform magnetic field. We exploit the C_{4v}-symmetry of the system in the
calculation of the correction terms. The numerical results for the
semiclassical values will be compared with values extracted from exact quantum
mechanical calculations. The comparison shows an excellent agreement and
demonstrates the power of the hbar expansion method. | nlin_CD |
Multiple Shooting Shadowing for Sensitivity Analysis of Chaotic
Dynamical Systems: Sensitivity analysis methods are important tools for research and design with
simulations. Many important simulations exhibit chaotic dynamics, including
scale-resolving turbulent fluid flow simulations. Unfortunately, conventional
sensitivity analysis methods are unable to compute useful gradient information
for long-time-averaged quantities in chaotic dynamical systems. Sensitivity
analysis with least squares shadowing (LSS) can compute useful gradient
information for a number of chaotic systems, including simulations of chaotic
vortex shedding and homogeneous isotropic turbulence. However, this gradient
information comes at a very high computational cost. This paper presents
multiple shooting shadowing (MSS), a more computationally efficient shadowing
approach than the original LSS approach. Through an analysis of the convergence
rate of MSS, it is shown that MSS can have lower memory usage and run time than
LSS. | nlin_CD |
Chaos Synchronization using Nonlinear Observers with applications to
Cryptography: The goal of this survey paper is to provide an introduction to chaos
synchronization using nonlinear observers and its applications in cryptography.
I start with an overview of cryptography. Then, I recall the basics of chaos
theory and how to use chaotic systems for cryptography, with an introduction to
the problem of chaos synchronization. Then, I present the theory of non-linear
observers, which is used for the synchronization of chaotic systems. I start
with an explanation of the observability problem. Then, I introduce some of the
classical observers: Kalman filter, Luenberger observer, Extended Kalman
filter, Thau's observer, and High gain observer. I finish by introducing the
more advanced observers: Adaptive observers, Unknown inputs observers, Sliding
mode observers and ANFIS (Adaptive Neuro-Fuzzy Inference Systems) observers. | nlin_CD |
Local Fractional Calculus: a Review: The purpose of this article is to review the developments related to the
notion of local fractional derivative introduced in 1996. We consider its
definition, properties, implications and possible applications. This involves
the local fractional Taylor expansion, Leibnitz rule, chain rule, etc. Among
applications we consider the local fractional diffusion equation for fractal
time processes and the relation between stress and strain for fractal media.
Finally, we indicate a stochastic version of local fractional differential
equation. | nlin_CD |
Complexity and non-separability of classical Liouvillian dynamics: We propose a simple complexity indicator of classical Liouvillian dynamics,
namely the separability entropy, which determines the logarithm of an effective
number of terms in a Schmidt decomposition of phase space density with respect
to an arbitrary fixed product basis. We show that linear growth of separability
entropy provides stricter criterion of complexity than Kolmogorov-Sinai
entropy, namely it requires that dynamics is exponentially unstable, non-linear
and non-markovian. | nlin_CD |
Synchronized states in chaotic systems coupled indirectly through a
dynamic environment: We consider synchronization of chaotic systems coupled indirectly through a
common environmnet where the environment has an intrinsic dynmics of its own
modulated via feedback from the systems. We find that a rich vareity of
synchronization behavior, such as in-phase, anti-phase,complete and anti-
synchronization is possible. We present an approximate stability analysis for
the different synchronization behaviors. The transitions to different states of
synchronous behaviour are analyzed in the parameter plane of coupling strengths
by numerical studies for specific cases such as Rossler and Lorenz systems and
are characterized using various indices such as correlation, average phase
difference and Lyapunov exponents. The threshold condition obtained from
numerical analysis is found to agree with that from the stability analysis. | nlin_CD |
Electronic implementation of a dynamical network with nearly identical
hybrid nodes via unstable dissipative systems: A circuit architecture is proposed and implemented for a dynamical network
composed of a type of hybrid chaotic oscillator based on Unstable Dissipative
Systems (UDS). The circuit architecture allows selecting a network topology
with its link attributes and to study, experimentally, the practical
synchronous collective behavior phenomena. Additionally, based on the theory of
dynamical networks, a mathematical model of the circuit was described, taking
into account the natural tolerance of the electronic components. The network is
analyzed both numerically and experimentally according to the parameters
mismatch between nodes. | nlin_CD |
Evolution of Rogue Waves in Interacting Wave Systems: Large amplitude water waves on deep water has long been known in the sea
faring community, and the cause of great concern for, e.g., oil platform
constructions. The concept of such freak waves is nowadays, thanks to satellite
and radar measurements, well established within the scientific community. There
are a number of important models and approaches for the theoretical description
of such waves. By analyzing the scaling behavior of freak wave formation in a
model of two interacting waves, described by two coupled nonlinear Schroedinger
equations, we show that there are two different dynamical scaling behaviors
above and below a critical angle theta_c of the direction of the interacting
waves below theta_c all wave systems evolve and display statistics similar to a
wave system of non-interacting waves. The results equally apply to other
systems described by the nonlinear Schroedinger equations, and should be of
interest when designing optical wave guides. | nlin_CD |
Critical bending point in the Lyapunov localization spectra of
many-particle systems: The localization spectra of Lyapunov vectors in many-particle systems at low
density exhibit a characteristic bending behavior. It is shown that this
behavior is due to a restriction on the maximum number of the most localized
Lyapunov vectors determined by the system configuration and mutual
orthogonality. For a quasi-one-dimensional system this leads to a predicted
bending point at n_c \approx 0.432 N for an N particle system. Numerical
evidence is presented that confirms this predicted bending point as a function
of the number of particles N. | nlin_CD |
Frequency spanning homoclinic families: A family of maps or flows depending on a parameter $\nu$ which varies in an
interval, spans a certain property if along the interval this property depends
continuously on the parameter and achieves some asymptotic values along it. We
consider families of periodically forced Hamiltonian systems for which the
appropriately scaled frequency $\bar{\omega}(\nu)$ is spanned, namely it covers
the semi-infinite line $[0,\infty).$ Under some natural assumptions on the
family of flows and its adiabatic limit, we construct a convenient labelling
scheme for the primary homoclinic orbits which may undergo a countable number
of bifurcations along this interval. Using this scheme we prove that a properly
defined flux function is $C^{1}$ in $\nu.$ Combining this proof with previous
results of RK and Poje, immediately establishes that the flux function and the
size of the chaotic zone depend on the frequency in a non-monotone fashion for
a large class of Hamiltonian flows. | nlin_CD |
Fluctuations and Transients in Quantum-Resonant Evolution: The quantum-resonant evolution of the mean kinetic energy (MKE) of the kicked
particle is studied in detail on different time scales for {\em general}
kicking potentials. It is shown that the asymptotic time behavior of a
wave-packet MKE is typically a linear growth with bounded fluctuations having a
simple number-theoretical origin. For a large class of wave packets, the MKE is
shown to be exactly the superposition of its asymptotic behavior and transient
logarithmic corrections. Both fluctuations and transients can be significant
for not too large times but they may vanish identically under some conditions.
In the case of incoherent mixtures of plane waves, it is shown that the MKE
never exhibits asymptotic fluctuations but transients usually occur. | nlin_CD |
Complex dynamics in two-dimensional coupling of quadratic maps: In the context of complex quadratic networks (CQNs) introduced previously, we
study escape radius and synchronization properties in two dimensional networks.
This establishing the first step towards more general results in
higher-dimensional networks. | nlin_CD |
Geometrical Models of the Phase Space Structures Governing Reaction
Dynamics: Hamiltonian dynamical systems possessing equilibria of ${saddle} \times
{centre} \times...\times {centre}$ stability type display \emph{reaction-type
dynamics} for energies close to the energy of such equilibria; entrance and
exit from certain regions of the phase space is only possible via narrow
\emph{bottlenecks} created by the influence of the equilibrium points. In this
paper we provide a thorough pedagogical description of the phase space
structures that are responsible for controlling transport in these problems. Of
central importance is the existence of a \emph{Normally Hyperbolic Invariant
Manifold (NHIM)}, whose \emph{stable and unstable manifolds} have sufficient
dimensionality to act as separatrices, partitioning energy surfaces into
regions of qualitatively distinct behavior. This NHIM forms the natural
(dynamical) equator of a (spherical) \emph{dividing surface} which locally
divides an energy surface into two components (`reactants' and `products'), one
on either side of the bottleneck. This dividing surface has all the desired
properties sought for in \emph{transition state theory} where reaction rates
are computed from the flux through a dividing surface. In fact, the dividing
surface that we construct is crossed exactly once by reactive trajectories, and
not crossed by nonreactive trajectories, and related to these properties,
minimizes the flux upon variation of the dividing surface.
We discuss three presentations of the energy surface and the phase space
structures contained in it for 2-degree-of-freedom (DoF) systems in the
threedimensional space $\R^3$, and two schematic models which capture many of
the essential features of the dynamics for $n$-DoF systems. In addition, we
elucidate the structure of the NHIM. | nlin_CD |
Field theory of the inverse cascade in two-dimensional turbulence: A two-dimensional fluid, stirred at high wavenumbers and damped by both
viscosity and linear friction, is modeled by a statistical field theory. The
fluid's long-distance behavior is studied using renormalization-group (RG)
methods, as begun by Forster, Nelson, and Stephen [Phys. Rev. A 16, 732
(1977)]. With friction, which dissipates energy at low wavenumbers, one expects
a stationary inverse energy cascade for strong enough stirring. While such
developed turbulence is beyond the quantitative reach of perturbation theory, a
combination of exact and perturbative results suggests a coherent picture of
the inverse cascade. The zero-friction fluctuation-dissipation theorem (FDT) is
derived from a generalized time-reversal symmetry and implies zero anomalous
dimension for the velocity even when friction is present. Thus the Kolmogorov
scaling of the inverse cascade cannot be explained by any RG fixed point. The
beta function for the dimensionless coupling ghat is computed through two
loops; the ghat^3 term is positive, as already known, but the ghat^5 term is
negative. An ideal cascade requires a linear beta function for large ghat,
consistent with a Pad\'e approximant to the Borel transform. The conjecture
that the Kolmogorov spectrum arises from an RG flow through large ghat is
compatible with other results, but the accurate k^{-5/3} scaling is not
explained and the Kolmogorov constant is not estimated. The lack of scale
invariance should produce intermittency in high-order structure functions, as
observed in some but not all numerical simulations of the inverse cascade. When
analogous RG methods are applied to the one-dimensional Burgers equation using
an FDT-preserving dimensional continuation, equipartition is obtained instead
of a cascade--in agreement with simulations. | nlin_CD |
Multiple-scale analysis and renormalization for pre-asymptotic scalar
transport: Pre-asymptotic transport of a scalar quantity passively advected by a
velocity field formed by a large-scale component superimposed to a small-scale
fluctuation is investigated both analytically and by means of numerical
simulations. Exploiting the multiple-scale expansion one arrives at a
Fokker--Planck equation which describes the pre-asymptotic scalar dynamics.
Such equation is associated to a Langevin equation involving a multiplicative
noise and an effective (compressible) drift. For the general case, no explicit
expression for both the effective drift and the effective diffusivity (actually
a tensorial field) can be obtained. We discuss an approximation under which an
explicit expression for the diffusivity (and thus for the drift) can be
obtained. Its expression permits to highlight the important fact that the
diffusivity explicitly depends on the large-scale advecting velocity. Finally,
the robustness of the aforementioned approximation is checked numerically by
means of direct numerical simulations. | nlin_CD |
Localization of Chaotic Resonance States due to a Partial Transport
Barrier: Chaotic eigenstates of quantum systems are known to localize on either side
of a classical partial transport barrier if the flux connecting the two sides
is quantum mechanically not resolved due to Heisenberg's uncertainty.
Surprisingly, in open systems with escape chaotic resonance states can localize
even if the flux is quantum mechanically resolved. We explain this using the
concept of conditionally invariant measures from classical dynamical systems by
introducing a new quantum mechanically relevant class of such fractal measures.
We numerically find quantum-to-classical correspondence for localization
transitions depending on the openness of the system and on the decay rate of
resonance states. | nlin_CD |
The Lyapunov dimension and its estimation via the Leonov method: Along with widely used numerical methods for estimating and computing the
Lyapunov dimension there is an effective analytical approach, proposed by G.A.
Leonov in 1991. The Leonov method is based on the direct Lyapunov method with
special Lyapunov-like functions. The advantage of this method is that it allows
one to estimate the Lyapunov dimension of invariant set without local- ization
of the set in the phase space and in many cases get effectively exact Lyapunov
dimension formula. In this survey the invariance of Lyapunov dimension with
respect to diffeomorphisms and its connection with the Leonov method are
discussed. An analog of Leonov method for discrete time dynamical systems is
suggested. In a simple but rigorous way, here it is presented the connection
between the Leonov method and the key related works in the area: by Kaplan and
Yorke (the concept of Lyapunov dimension, 1979), Douady and Oesterle (upper
bounds of Hausdorff dimension via the Lyapunov dimension of maps, 1980),
Constantin, Eden, Foias, and Temam (upper bounds of Hausdorff dimension via the
Lyapunov exponents and dimension of dynamical systems, 1985-90), and the
numerical calculation of the Lyapunov exponents and dimension. | nlin_CD |
Robust chaos with variable Lyapunov exponent in smooth one-dimensional
maps: We present several new easy ways of generating smooth one-dimensional maps
displaying robust chaos, i.e., chaos for whole intervals of the parameter.
Unlike what happens with previous methods, the Lyapunov exponent of the maps
constructed here varies widely with the parameter. We show that the condition
of negative Schwarzian derivative, which was used in previous works, is not a
necessary condition for robust chaos. Finally we show that the maps constructed
in previous works have always the Lyapunov exponent $\ln 2$ because they are
conjugated to each other and to the tent map by means of smooth homeomorphisms.
In the methods presented here, the maps have variable Lyapunov coefficients
because they are conjugated through non-smooth homeomorphisms similar to
Minkowski's question mark function. | nlin_CD |
Modulated amplitude waves with nonzero phases in Bose-Einstein
condensates: In this paper we give a frame for application of the averaging method to
Bose-Einstein condensates (BECs) and obtain an abstract result upon the
dynamics of BECs. Using aver- aging method, we determine the location where the
modulated amplitude waves (periodic or quasi-periodic) exist and we also study
the stability and instability of modulated amplitude waves (periodic or
quasi-periodic). Compared with the previous work, modulated amplitude waves
studied in this paper have nontrivial phases and this makes the problem become
more diffcult, since it involves some singularities. | nlin_CD |
Intermittent generalized synchronization in unidirectionally coupled
chaotic oscillators: A new behavior type of unidirectionally coupled chaotic oscillators near the
generalized synchronization transition has been detected. It has been shown
that the generalized synchronization appearance is preceded by the intermitted
behavior: close to threshold parameter value the coupled chaotic systems
demonstrate the generalized synchronization most of the time, but there are
time intervals during which the synchronized oscillations are interrupted by
non-synchronous bursts. This type of the system behavior has been called
intermitted generalized synchronization (IGS) by analogy with intermitted lag
synchronization (ILS) [Phys. Rev. E \textbf{62}, 7497 (2000)]. | nlin_CD |
Mass fluctuations and diffusion in time-dependent random environments: A mass ejection model in a time-dependent random environment with both
temporal and spatial correlations is introduced. When the environment has a
finite correlation length, individual particle trajectories are found to
diffuse at large times with a displacement distribution that approaches a
Gaussian. The collective dynamics of diffusing particles reaches a
statistically stationary state, which is characterized in terms of a
fluctuating mass density field. The probability distribution of density is
studied numerically for both smooth and non-smooth scale-invariant random
environments. A competition between trapping in the regions where the ejection
rate of the environment vanishes and mixing due to its temporal dependence
leads to large fluctuations of mass. These mechanisms are found to result in
the presence of intermediate power-law tails in the probability distribution of
the mass density. For spatially differentiable environments, the exponent of
the right tail is shown to be universal and equal to -3/2. However, at small
values, it is found to depend on the environment. Finally, spatial scaling
properties of the mass distribution are investigated. The distribution of the
coarse-grained density is shown to posses some rescaling properties that depend
on the scale, the amplitude of the ejection rate, and the H\"older exponent of
the environment. | nlin_CD |
The origin of diffusion: the case of non chaotic systems: We investigate the origin of diffusion in non-chaotic systems. As an example,
we consider 1-$d$ map models whose slope is everywhere 1 (therefore the
Lyapunov exponent is zero) but with random quenched discontinuities and
quasi-periodic forcing. The models are constructed as non-chaotic
approximations of chaotic maps showing deterministic diffusion, and represent
one-dimensional versions of a Lorentz gas with polygonal obstacles (e.g., the
Ehrenfest wind tree model). In particular, a simple construction shows that
these maps define non-chaotic billiards in space-time. The models exhibit, in a
wide range of the parameters, the same diffusive behavior of the corresponding
chaotic versions. We present evidence of two sufficient ingredients for
diffusive behavior in one-dimensional, non-chaotic systems: i) a finite-size,
algebraic instability mechanism, and ii) a mechanism that suppresses periodic
orbits. | nlin_CD |
The Kuramoto Model with Time-Varying Parameters: We introduce a generalization of the Kuramoto model by explicit consideration
of time-dependent parameters. The oscillators' natural frequencies and/or
couplings are supposed to be influenced by external, time-dependant fields,
with constant or randomly-distributed strengths. As a result, the dynamics of
an external system is being imposed on top of the autonomous one, a scenario
that cannot be treated adequately by previous (adiabatic) approaches. We now
propose an analysis which describes faithfully the overall dynamics of the
system. | nlin_CD |
A mechanical model of normal and anomalous diffusion: The overdamped dynamics of a charged particle driven by an uniform electric
field through a random sequence of scatterers in one dimension is investigated.
Analytic expressions of the mean velocity and of the velocity power spectrum
are presented. These show that above a threshold value of the field normal
diffusion is superimposed to ballistic motion. The diffusion constant can be
given explicitly. At the threshold field the transition between conduction and
localization is accompanied by an anomalous diffusion. Our results exemplify
that, even in the absence of time-dependent stochastic forces, a purely
mechanical model equipped with a quenched disorder can exhibit normal as well
as anomalous diffusion, the latter emerging as a critical property. | nlin_CD |
Detection of synchronization from univariate data using wavelet
transform: A method is proposed for detecting from univariate data the presence of
synchronization of a self-sustained oscillator by external driving with varying
frequency. The method is based on the analysis of difference between the
oscillator instantaneous phases calculated using continuous wavelet transform
at time moments shifted by a certain constant value relative to each other. We
apply our method to a driven asymmetric van der Pol oscillator, experimental
data from a driven electronic oscillator with delayed feedback and human
heartbeat time series. In the latest case, the analysis of the heart rate
variability data reveals synchronous regimes between the respiration and slow
oscillations in blood pressure. | nlin_CD |
The dynamics of a driven harmonic oscillator coupled to independent
Ising spins in random fields: We aim at an understanding of the dynamical properties of a periodically
driven damped harmonic oscillator coupled to a Random Field Ising Model (RFIM)
at zero temperature, which is capable to show complex hysteresis. The system is
a combination of a continuous (harmonic oscillator) and a discrete (RFIM)
subsystem, which classifies it as a hybrid system. In this paper we focus on
the hybrid nature of the system and consider only independent spins in quenched
random local fields, which can already lead to complex dynamics such as chaos
and multistability. We study the dynamic behavior of this system by using the
theory of piecewise-smooth dynamical systems and discontinuity mappings.
Specifically, we present bifurcation diagrams, Lyapunov exponents as well as
results for the shape and the dimensions of the attractors and the
self-averaging behavior of the attractor dimensions and the magnetization.
Furthermore we investigate the dynamical behavior of the system for an
increasing number of spins and the transition to the thermodynamic limit, where
the system behaves like a driven harmonic oscillator with an additional
nonlinear smooth external force. | nlin_CD |
Thermalization of Classical Weakly Nonintegrable Many-Body Systems: We devote our studies to the subject of weakly nonintegrable dynamics of
systems with a macroscopic number of degrees of freedom. Our main points of
interest are the relations between the timescales of thermalization and the
timescales of chaotization; the choice of appropriate observables and the
structure of equations coupling them; identifying the classes of weakly
nonintegrable dynamics and developing tools to diagnose the properties specific
to such classes. We discuss the traditional in the field methods for
thermalization timescale computation and employ them to study the scaling the
timescale with the proximity to the integrable limit. We then elaborate on a
novel framework based on the full Lyapunov spectra computation for large
systems as a powerful tool for the characterization of weak nonintegrability.
In particular, the Lyapunov spectrum scaling offers a quantitative description
allowing us to infer the structure of the underlying network of observables.
Proximity to integrable limit is associated with the rapid growth of
thermalization timescales and, thus, potential numerical challenges. We solve
these challenges by performing numerical tests using computationally efficient
model - unitary maps. The great advantage of unitary maps for numerical
applications is time-discrete error-free evolution. We use these advantages to
perform large timescale and system size computations in extreme proximity to
the integrable limit. To demonstrate the scope of obtained results we report on
the application of the developed framework to Hamiltonian systems. | nlin_CD |
Sketching 1-D stable manifolds of 2-D maps without the inverse: Saddle fixed points are the centerpieces of complicated dynamics in a system.
The one-dimensional stable and unstable manifolds of these saddle-points are
crucial to understanding the dynamics of such systems. While the problem of
sketching the unstable manifold is simple, plotting the stable manifold is not
as easy. Several algorithms exist to compute the stable manifold of
saddle-points, but they have their limitations, especially when the system is
not invertible. In this paper, we present a new algorithm to compute the stable
manifold of 2-dimensional systems which can also be used for non-invertible
systems. After outlining the logic of the algorithm, we demonstrate the output
of the algorithm on several examples. | nlin_CD |
Properties of maximum Lempel-Ziv complexity strings: The properties of maximum Lempel-Ziv complexity strings are studied for the
binary case. A comparison between MLZs and random strings is carried out. The
length profile of both type of sequences show different distribution functions.
The non-stationary character of the MLZs are discussed. The issue of
sensitiveness to noise is also addressed. An empirical ansatz is found that
fits well to the Lempel-Ziv complexity of the MLZs for all lengths up to $10^6$
symbols. | nlin_CD |
Numerical investigation on the Hill's type lunar problem with
homogeneous potential: We consider the planar Hill's lunar problem with a homogeneous gravitational
potential. The investigation of the system is twofold. First, the starting
conditions of the trajectories are classified into three classes, that is
bounded, escaping, and collisional. Second, we study the no-return property of
the Lagrange point $L_2$ and we observe that the escaping trajectories are
scattered exponentially. Moreover, it is seen that in the supercritical case,
with $\alpha \geq 2$, the basin boundaries are smooth. On the other hand, in
the subcritical case, with $\alpha < 2$ the boundaries between the different
types of basins exhibit fractal properties. | nlin_CD |
The evolution of anisotropic structures and turbulence in the
multi-dimensional Burgers equation: The goal of the present paper is the investigation of the evolution of
anisotropic regular structures and turbulence at large Reynolds number in the
multi-dimensional Burgers equation. We show that we have local isotropization
of the velocity and potential fields at small scale inside cellular zones. For
periodic waves, we have simple decay inside of a frozen structure. The global
structure at large times is determined by the initial correlations, and for
short range correlated field, we have isotropization of turbulence. The other
limit we consider is the final behavior of the field, when the processes of
nonlinear and harmonic interactions are frozen, and the evolution of the field
is determined only by the linear dissipation. | nlin_CD |
Anomalous Scaling on a Spatiotemporally Chaotic Attractor: The Nikolaevskiy model for pattern formation with continuous symmetry
exhibits spatiotemporal chaos with strong scale separation. Extensive numerical
investigations of the chaotic attractor reveal unexpected scaling behavior of
the long-wave modes. Surprisingly, the computed amplitude and correlation time
scalings are found to differ from the values obtained by asymptotically
consistent multiple-scale analysis. However, when higher-order corrections are
added to the leading-order theory of Matthews and Cox, the anomalous scaling is
recovered. | nlin_CD |
Robust Approach for Rotor Mapping in Cardiac Tissue: The motion of and interaction between phase singularities that anchor spiral
waves captures many qualitative and, in some cases, quantitative features of
complex dynamics in excitable systems. Being able to accurately reconstruct
their position is thus quite important, even if the data are noisy and sparse,
as in electrophysiology studies of cardiac arrhythmias, for instance. A
recently proposed global topological approach [Marcotte & Grigoriev, Chaos 27,
093936 (2017)] promises to dramatically improve the quality of the
reconstruction compared with traditional, local approaches. Indeed, we found
that this approach is capable of handling noise levels exceeding the range of
the signal with minimal loss of accuracy. Moreover, it also works successfully
with data sampled on sparse grids with spacing comparable to the mean
separation between the phase singularities for complex patterns featuring
multiple interacting spiral waves. | nlin_CD |
Amplitude death and resurgence of oscillation in network of mobile
oscillators: The phenomenon of amplitude death has been explored using a variety of
different coupling strategies in the last two decades. In most of the work, the
basic coupling arrangement is considered to be static over time, although many
realistic systems exhibit significant changes in the interaction pattern as
time varies. In this article, we study the emergence of amplitude death in a
dynamical network composed of time-varying interaction amidst a collection of
random walkers in a finite region of three dimensional space. We consider an
oscillator for each walker and demonstrate that depending upon the network
parameters and hence the interaction between them, global oscillation in the
network gets suppressed. In this framework, vision range of each oscillator
decides the number of oscillators with which it interacts. In addition, with
the use of an appropriate feedback parameter in the coupling strategy, we
articulate how the suppressed oscillation can be resurrected in the systems'
parameter space. The phenomenon of amplitude death and the resurgence of
oscillation is investigated taking limit cycle and chaotic oscillators for
broad ranges of parameters, like interaction strength k between the entities,
vision range r and the speed of movement v. | nlin_CD |
Can recurrence networks show small world property?: Recurrence networks are complex networks, constructed from time series data,
having several practical applications. Though their properties when constructed
with the threshold value \epsilon chosen at or just above the percolation
threshold of the network are quite well understood, what happens as the
threshold increases beyond the usual operational window is still not clear from
a complex network perspective. The present Letter is focused mainly on the
network properties at intermediate-to-large values of the recurrence threshold,
for which no systematic study has been performed so far. We argue, with
numerical support, that recurrence networks constructed from chaotic attractors
with \epsilon equal to the usual recurrence threshold or slightly above cannot,
in general, show small-world property. However, if the threshold is further
increased, the recurrence network topology initially changes to a
small-worldstructure and finally to that of a classical random graph as the
threshold approaches the size of the strange attractor. | nlin_CD |
Synchronization of oscillators with hyperbolic chaotic phases: Synchronization in a population of oscillators with hyperbolic chaotic phases
is studied for two models. One is based on the Kuramoto dynamics of the phase
oscillators and on the Bernoulli map applied to these phases. This system
possesses an Ott-Antonsen invariant manifold, allowing for a derivation of a
map for the evolution of the complex order parameter. Beyond a critical
coupling strength, this model demonstrates bistability synchrony-disorder.
Another model is based on the coupled autonomous oscillators with hyperbolic
chaotic strange attractors of Smale-Williams type. Here a disordered
asynchronous state at small coupling strengths, and a completely synchronous
state at large couplings are observed. Intermediate regimes are characterized
by different levels of complexity of the global order parameter dynamics. | nlin_CD |
Dirac comb and exponential frequency spectra in chaos and nonlinear
dynamics: An exponential frequency power spectral density is a well known property of
many continuous time chaotic systems and has been attributed to the presence of
Lorentzian-shaped pulses in the time series of the dynamical variables. Here a
stochastic model of such fluctuations is presented, describing these as a
super-position of pulses with fixed shape and constant duration. Closed form
expressions are derived for the lowest order moments, auto-correlation function
and frequency power spectral density in the case of periodic pulse arrivals and
a random distribution of pulse amplitudes. In general, the spectrum is a Dirac
comb located at multiples of the inverse periodicity time and modulated by the
pulse spectrum. For Lorentzian-shaped pulses there is an exponential modulation
of the Dirac comb. Deviations from strict periodicity in the arrivals
efficiently removes the Dirac comb, leaving only the spectrum of the pulse
function. This effect is also achieved if the pulse amplitudes are independent
of the arrivals and have vanishing mean value. Randomness in the pulse arrival
times is investigated by numerical realizations of the process, and the model
is used to describe the power spectral densities of time series from the Lorenz
system. | nlin_CD |
Fractional Dissipative Standard Map: Using kicked differential equations of motion with derivatives of noninteger
orders, we obtain generalizations of the dissipative standard map. The main
property of these generalized maps, which are called fractional maps, is
long-term memory.The memory effect in the fractional maps means that their
present state of evolution depends on all past states with special forms of
weights. Already a small deviation of the order of derivative from the integer
value corresponding to the regular dissipative standard map (small memory
effects) leads to the qualitatively new behavior of the corresponding
attractors. The fractional dissipative standard maps are used to demonstrate a
new type of fractional attractors in the wide range of the fractional orders of
derivatives. | nlin_CD |
A new deterministic model for chaotic reversals: We present a new chaotic system of three coupled ordinary differential
equations, limited to quadratic nonlinear terms. A wide variety of dynamical
regimes are reported. For some parameters, chaotic reversals of the amplitudes
are produced by crisis-induced intermittency, following a mechanism different
from what is generally observed in similar deterministic models. Despite its
simplicity, this system therefore generates a rich dynamics, able to model more
complex physical systems. In particular, a comparison with reversals of the
magnetic field of the Earth shows a surprisingly good agreement, and highlights
the relevance of deterministic chaos to describe geomagnetic field dynamics. | nlin_CD |
Decomposing the Dynamics of the Lorenz 1963 model using Unstable
Periodic Orbits: Averages, Transitions, and Quasi-Invariant Sets: Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic
dynamical systems, as they allow one to distill their dynamical structure. We
consider here the Lorenz 1963 model with the classic parameters' value. We
investigate how a chaotic trajectory can be approximated using a complete set
of UPOs up to symbolic dynamics' period 14. At each instant, we rank the UPOs
according to their proximity to the position of the orbit in the phase space.
We study this process from two different perspectives. First, we find that
longer period UPOs overwhelmingly provide the best local approximation to the
trajectory. Second, we construct a finite-state Markov chain by studying the
scattering of the orbit between the neighbourhood of the various UPOs. Each UPO
and its neighbourhood are taken as a possible state of the system. Through the
analysis of the subdominant eigenvectors of the corresponding stochastic matrix
we provide a different interpretation of the mixing processes occurring in the
system by taking advantage of the concept of quasi-invariant sets. | nlin_CD |
Nodal Domain Statistics for Quantum Maps, Percolation and SLE: We develop a percolation model for nodal domains in the eigenvectors of
quantum chaotic torus maps. Our model follows directly from the assumption that
the quantum maps are described by random matrix theory. Its accuracy in
predicting statistical properties of the nodal domains is demonstrated by
numerical computations for perturbed cat maps and supports the use of
percolation theory to describe the wave functions of general hamiltonian
systems, where the validity of the underlying assumptions is much less clear.
We also demonstrate that the nodal domains of the perturbed cat maps obey the
Cardy crossing formula and find evidence that the boundaries of the nodal
domains are described by SLE with $\kappa$ close to the expected value of 6,
suggesting that quantum chaotic wave functions may exhibit conformal invariance
in the semiclassical limit. | nlin_CD |
Fluctuational transitions through a fractal basin boundary: Fluctuational transitions between two co-existing chaotic attractors,
separated by a fractal basin boundary, are studied in a discrete dynamical
system. It is shown that the mechanism for such transitions is determined by a
hierarchy of homoclinic points. The most probable escape path from the chaotic
attractor to the fractal boundary is found using both statistical analyses of
fluctuational trajectories and the Hamiltonian theory of fluctuations. | nlin_CD |
Directed transport and Floquet analysis for a periodically kicked
wavepacket at a quantum resonance: The dynamics of a kicked quantum mechanical wavepacket at a quantum resonance
is studied in the framework of Floquet analysis. It is seen how a directed
current can be created out of a homogeneous initial state at certain resonances
in an asymmetric potential. The almost periodic parameter dependence of the
current is found to be connected with level crossings in the Floquet spectrum. | nlin_CD |
Low-frequency variability and heat transport in a low-order nonlinear
coupled ocean-atmosphere model: We formulate and study a low-order nonlinear coupled ocean-atmosphere model
with an emphasis on the impact of radiative and heat fluxes and of the
frictional coupling between the two components. This model version extends a
previous 24-variable version by adding a dynamical equation for the passive
advection of temperature in the ocean, together with an energy balance model.
The bifurcation analysis and the numerical integration of the model reveal
the presence of low-frequency variability (LFV) concentrated on and near a
long-periodic, attracting orbit. This orbit combines atmospheric and oceanic
modes, and it arises for large values of the meridional gradient of radiative
input and of frictional coupling. Chaotic behavior develops around this orbit
as it loses its stability; this behavior is still dominated by the LFV on
decadal and multi-decadal time scales that is typical of oceanic processes.
Atmospheric diagnostics also reveals the presence of predominant low- and
high-pressure zones, as well as of a subtropical jet; these features recall
realistic climatological properties of the oceanic atmosphere.
Finally, a predictability analysis is performed. Once the decadal-scale
periodic orbits develop, the coupled system's short-term instabilities --- as
measured by its Lyapunov exponents --- are drastically reduced, indicating the
ocean's stabilizing role on the atmospheric dynamics. On decadal time scales,
the recurrence of the solution in a certain region of the invariant subspace
associated with slow modes displays some extended predictability, as reflected
by the oscillatory behavior of the error for the atmospheric variables at long
lead times. | nlin_CD |
Semiclassical cross section correlations: We calculate within a semiclassical approximation the autocorrelation
function of cross sections. The starting point is the semiclassical expression
for the diagonal matrix elements of an operator. For general operators with a
smooth classical limit the autocorrelation function of such matrix elements has
two contributions with relative weights determined by classical dynamics. We
show how the random matrix result can be obtained if the operator approaches a
projector onto a single initial state. The expressions are verified in
calculations for the kicked rotor. | nlin_CD |
Scaling of Chaos in Strongly Nonlinear Lattices: Although it is now understood that chaos in complex classical systems is the
foundation of thermodynamic behavior, the detailed relations between the
microscopic properties of the chaotic dynamics and the macroscopic
thermodynamic observations still remain mostly in the dark. In this work, we
numerically analyze the probability of chaos in strongly nonlinear Hamiltonian
systems and find different scaling properties depending on the nonlinear
structure of the model. We argue that these different scaling laws of chaos
have definite consequences for the macroscopic diffusive behavior, as chaos is
the microscopic mechanism of diffusion. This is compared with previous results
on chaotic diffusion [New J.\ Phys.\ 15, 053015 (2013)], and a relation between
microscopic chaos and macroscopic diffusion is established. | nlin_CD |
Quantum-classical correspondence for local density of states and
eigenfunctions of a chaotic periodic billiard: Classical-quantum correspondence for conservative chaotic Hamiltonians is
investigated in terms of the structure of the eigenfunctions and the local
density of states, using as a model a 2D rippled billiard in the regime of
global chaos. The influence of the observed localized and sparsed states in the
quantum-classical correspondence is discussed. | nlin_CD |
Multifractal concentrations of inertial particles in smooth random flows: Collisionless suspensions of inertial particles (finite-size impurities) are
studied in 2D and 3D spatially smooth flows. Tools borrowed from the study of
random dynamical systems are used to identify and to characterise in full
generality the mechanisms leading to the formation of strong inhomogeneities in
the particle concentration.
Phenomenological arguments are used to show that in 2D, heavy particles form
dynamical fractal clusters when their Stokes number (non-dimensional viscous
friction time) is below some critical value. Numerical simulations provide
strong evidence for this threshold in both 2D and 3D and for particles not only
heavier but also lighter than the carrier fluid. In 2D, light particles are
found to cluster at discrete (time-dependent) positions and velocities in some
range of the dynamical parameters (the Stokes number and the mass density ratio
between fluid and particles). This regime is absent in 3D, where evidence is
that the Hausdorff dimension of clusters in phase space (position-velocity)
remains always above two.
After relaxation of transients, the phase-space density of particles becomes
a singular random measure with non-trivial multiscaling properties. Theoretical
results about the projection of fractal sets are used to relate the
distribution in phase space to the distribution of the particle positions.
Multifractality in phase space implies also multiscaling of the spatial
distribution of the mass of particles. Two-dimensional simulations, using
simple random flows and heavy particles, allow the accurate determination of
the scaling exponents: anomalous deviations from self-similar scaling are
already observed for Stokes numbers as small as $10^{-4}$. | nlin_CD |
Physics-Enhanced Bifurcation Optimisers: All You Need Is a Canonical
Complex Network: Many physical systems with the dynamical evolution that at its steady state
gives a solution to optimization problems were proposed and realized as
promising alternatives to conventional computing. Systems of oscillators such
as coherent Ising and XY machines based on lasers, optical parametric
oscillators, memristors, polariton and photon condensates are particularly
promising due to their scalability, low power consumption and room temperature
operation. They achieve a solution via the bifurcation of the fundamental
supermode that globally minimizes either the power dissipation of the system or
the system Hamiltonian. We show that the canonical Andronov-Hopf networks can
capture the bifurcation behaviour of the physical optimizer. Furthermore, a
continuous change of variables transforms any physical optimizer into the
canonical network so that the success of the physical XY-Ising machine depends
primarily on how well the parameters of the networks can be controlled. Our
work, therefore, places different physical optimizers in the same mathematical
framework that allows for the hybridization of ideas across disparate physical
platforms. | nlin_CD |
Predictors and Predictands of Linear Response in Spatially Extended
Systems: The goal of response theory, in each of its many statistical mechanical
formulations, is to predict the perturbed response of a system from the
knowledge of the unperturbed state and of the applied perturbation. A new
recent angle on the problem focuses on providing a method to perform
predictions of the change in one observable of the system by using the change
in a second observable as a surrogate for the actual forcing. Such a viewpoint
tries to address the very relevant problem of causal links within complex
system when only incomplete information is available. We present here a method
for quantifying and ranking the predictive ability of observables and use it to
investigate the response of a paradigmatic spatially extended system, the
Lorenz '96 model. We perturb locally the system and we then study to what
extent a given local observable can predict the behaviour of a separate local
observable. We show that this approach can reveal insights on the way a signal
propagates inside the system. We also show that the procedure becomes more
efficient if one considers multiple acting forcings and, correspondingly,
multiple observables as predictors of the observable of interest. | nlin_CD |
Model error and sequential data assimilation. A deterministic
formulation: Data assimilation schemes are confronted with the presence of model errors
arising from the imperfect description of atmospheric dynamics. These errors
are usually modeled on the basis of simple assumptions such as bias, white
noise, first order Markov process. In the present work, a formulation of the
sequential extended Kalman filter is proposed, based on recent findings on the
universal deterministic behavior of model errors in deep contrast with previous
approaches (Nicolis, 2004). This new scheme is applied in the context of a
spatially distributed system proposed by Lorenz (1996). It is found that (i)
for short times, the estimation error is accurately approximated by an
evolution law in which the variance of the model error (assumed to be a
deterministic process) evolves according to a quadratic law, in agreement with
the theory. Moreover, the correlation with the initial condition error appears
to play a secondary role in the short time dynamics of the estimation error
covariance. (ii) The deterministic description of the model error evolution,
incorporated into the classical extended Kalman filter equations, reveals that
substantial improvements of the filter accuracy can be gained as compared with
the classical white noise assumption. The universal, short time, quadratic law
for the evolution of the model error covariance matrix seems very promising for
modeling estimation error dynamics in sequential data assimilation. | nlin_CD |
Statistics of surface gravity wave turbulence in the space and time
domains: We present experimental results on simultaneous space-time measurements for
the gravity wave turbulence in a large laboratory flume. We compare these
results with predictions of the weak turbulence theory (WTT) based on random
waves, as well as with predictions based on the coherent singular wave crests.
We see that both wavenumber and the frequency spectra are not universal and
dependent on the wave strength, with some evidence in favor of WTT at larger
wave intensities when the finite flume effects are minimal. We present further
theoretical analysis of the role of the random and coherent waves in the wave
probability density function (PDF) and the structure functions (SFs). Analyzing
our experimental data we found that the random waves and the coherent
structures/breaks coexist: the former show themselves in a quasi-gaussian PDF
core and in the low-order SFs, and the latter - in the PDF tails and the
high-order SF's. It appears that the x-space signal is more intermittent than
the t-space signal, and the x-space SFs capture more singular coherent
structures than do the t-space SFs. We outline an approach treating the
interactions of these random and coherent components as a turbulence cycle
characterized by the turbulence fluxes in both the wavenumber and the amplitude
spaces. | nlin_CD |
Dynamical localization in kicked rotator as a paradigm of other systems:
spectral statistics and the localization measure: We study the intermediate statistics of the spectrum of quasi-energies and of
the eigenfunctions in the kicked rotator, in the case when the corresponding
system is fully chaotic while quantally localized. As for the eigenphases, we
find clear evidence that the spectral statistics is well described by the Brody
distribution, notably better than by the Izrailev's one, which has been
proposed and used broadly to describe such cases. We also studied the
eigenfunctions of the Floquet operator and their localization. We show the
existence of a scaling law between the repulsion parameter with relative
localization length, but only as a first order approximation, since another
parameter plays a role. We believe and have evidence that a similar analysis
applies in time-independent Hamilton systems. | nlin_CD |
Effect of Noise on the Standard Mapping: The effect of a small amount of noise on the standard mapping is considered.
Whenever the standard mapping possesses accelerator modes (where the action
increases approximately linearly with time), the diffusion coefficient contains
a term proportional to the reciprocal of the variance of the noise term. At
large values of the stochasticity parameter, the accelerator modes exhibit a
universal behavior. As a result the dependence of the diffusion coefficient on
the stochasticity parameter also shows some universal behavior. | nlin_CD |
Bubble doubling route to strange nonchaotic attractor in a
quasiperiodically forced Chua's circuit: We have identified a novel mechanism for the birth of Strange Nonchaotic
Attractor (SNA) in a quasiperiodically forced Chua's circuit. In this study the
amplitude of one of the external driving forces is considered as the control
parameter. By varying this control parameter, we find that bubbles appear in
the strands of the torus. These bubbles start to double in number as the
control parameter is increased. On increasing the parameter continuously,
successive doubling of the bubbles occurs, leading to the birth of SNAs. We
call this mechanism as the bubble doubling mechanism. The formation of SNA
through this bubble doubling route is confirmed numerically, using Poincar\'e
maps, maximal Lyapunov exponent and its variance and the distribution of
finite-time Lyapunov exponents. Also a quantitative confirmation of the strange
nonchaotic dynamics is carried out with the help of singular continuous
spectrum analysis. | nlin_CD |
The semiclassical relation between open trajectories and periodic orbits
for the Wigner time delay: The Wigner time delay of a classically chaotic quantum system can be
expressed semiclassically either in terms of pairs of scattering trajectories
that enter and leave the system or in terms of the periodic orbits trapped
inside the system. We show how these two pictures are related on the
semiclassical level. We start from the semiclassical formula with the
scattering trajectories and derive from it all terms in the periodic orbit
formula for the time delay. The main ingredient in this calculation is a new
type of correlation between scattering trajectories which is due to
trajectories that approach the trapped periodic orbits closely. The equivalence
between the two pictures is also demonstrated by considering correlation
functions of the time delay. A corresponding calculation for the conductance
gives no periodic orbit contributions in leading order. | nlin_CD |
Delay time modulation induced oscillating synchronization and
intermittent anticipatory/lag and complete synchronizations in time-delay
nonlinear dynamical systems: Existence of a new type of oscillating synchronization that oscillates
between three different types of synchronizations (anticipatory, complete and
lag synchronizations) is identified in unidirectionally coupled nonlinear
time-delay systems having two different time-delays, that is feedback delay
with a periodic delay time modulation and a constant coupling delay.
Intermittent anticipatory, intermittent lag and complete synchronizations are
shown to exist in the same system with identical delay time modulations in both
the delays. The transition from anticipatory to complete synchronization and
from complete to lag synchronization as a function of coupling delay with
suitable stability condition is discussed. The intermittent anticipatory and
lag synchronizations are characterized by the minimum of similarity functions
and the intermittent behavior is characterized by a universal asymptotic
$-{3/2}$ power law distribution. It is also shown that the delay time carved
out of the trajectories of the time-delay system with periodic delay time
modulation cannot be estimated using conventional methods, thereby reducing the
possibility of decoding the message by phase space reconstruction. | nlin_CD |
Freely decaying weak turbulence for sea surface gravity waves: We study numerically the generation of power laws in the framework of weak
turbulence theory for surface gravity waves in deep water. Starting from a
random wave field, we let the system evolve numerically according to the
nonlinear Euler equations for gravity waves in infinitely deep water. In
agreement with the theory of Zakharov and Filonenko, we find the formation of a
power spectrum characterized by a power law of the form of $|{\bf k}|^{-2.5}$. | nlin_CD |
Study of wave chaos in a randomly-inhomogeneous oceanic acoustic
waveguide: spectral analysis of the finite-range evolution operator: The proplem of sound propagation in an oceanic waveguide is considered.
Scattering on random inhomogeneity of the waveguide leads to wave chaos. Chaos
reveals itself in spectral properties of the finite-range evolution operator
(FREO). FREO describes transformation of a wavefield in course of propagation
along a finite segment of a waveguide. We study transition to chaos by tracking
variations in spectral statistics with increasing length of the segment.
Analysis of the FREO is accompanied with ray calculations using the one-step
Poincar\'e map which is the classical counterpart of the FREO. Underwater sound
channel in the Sea of Japan is taken for an example. Several methods of
spectral analysis are utilized. In particular, we approximate level spacing
statistics by means of the Berry-Robnik and Brody distributions, explore the
spectrum using the procedure elaborated by A. Relano with coworkers (Relano et
al, Phys. Rev. Lett., 2002; Relano, Phys. Rev. Lett., 2008), and analyze modal
expansions of the eigenfunctions. We show that the analysis of FREO
eigenfunctions is more informative than the analysis of eigenvalue statistics.
It is found that near-axial sound propagation in the Sea of Japan preserves
stability even over distances of hundreds kilometers. This phenomenon is
associated with the presence of a shearless torus in the classical phase space.
Increasing of acoustic wavelength degrades scattering, resulting in recovery of
localization near periodic orbits of the one-step Poincar\'e map. Relying upon
the formal analogy between wave and quantum chaos, we suggest that the concept
of FREO, supported by classical calculations via the one-step Poincar\'e map,
can be efficiently applied for studying chaos-induced decoherence in quantum
systems. | nlin_CD |
Recovery of chaotic tunneling due to destruction of dynamical
localization by external noise: Quantum tunneling in the presence of chaos is analyzed, focusing especially
on the interplay between quantum tunneling and dynamical localization. We
observed flooding of potentially existing tunneling amplitude by adding noise
to the chaotic sea to attenuate the destructive interference generating
dynamical localization. This phenomenon is related to the nature of complex
orbits describing tunneling between torus and chaotic regions. The tunneling
rate is found to obey a perturbative scaling with noise intensity when the
noise intensity is sufficiently small and then saturate in a large noise
intensity regime. A relation between the tunneling rate and the localization
length of the chaotic states is also demonstrated. It is shown that due to the
competition between dynamical tunneling and dynamical localization, the
tunneling rate is not a monotonically increasing function of Planck's constant.
The above results are obtained for a system with a sharp border between torus
and chaotic regions. The validity of the results for a system with a smoothed
border is also explained. | nlin_CD |
Geometry of complex instability and escape in four-dimensional
symplectic maps: In four-dimensional symplectic maps complex instability of periodic orbits is
possible, which cannot occur in the two-dimensional case. We investigate the
transition from stable to complex unstable dynamics of a fixed point under
parameter variation. The change in the geometry of regular structures is
visualized using 3D phase-space slices and in frequency space using the example
of two coupled standard maps. The chaotic dynamics is studied using escape time
plots and by computations of the 2D invariant manifolds associated with the
complex unstable fixed point. Based on a normal-form description, we
investigate the underlying transport mechanism by visualizing the escape paths
and the long-time confinement in the surrounding of the complex unstable fixed
point. We find that the escape is governed by the transport along the unstable
manifold across invariant planes of the normal-form. | nlin_CD |
Fractal structures of normal and anomalous diffusion in nonlinear
nonhyperbolic dynamical systems: A paradigmatic nonhyperbolic dynamical system exhibiting deterministic
diffusion is the smooth nonlinear climbing sine map. We find that this map
generates fractal hierarchies of normal and anomalous diffusive regions as
functions of the control parameter. The measure of these self-similar sets is
positive, parameter-dependent, and in case of normal diffusion it shows a
fractal diffusion coefficient. By using a Green-Kubo formula we link these
fractal structures to the nonlinear microscopic dynamics in terms of fractal
Takagi-like functions. | nlin_CD |
Large coupled oscillator systems with heterogeneous interaction delays: In order to discover generic effects of heterogeneous communication delays on
the dynamics of large systems of coupled oscillators, this paper studies a
modification of the Kuramoto model incorporating a distribution of interaction
delays. By focusing attention on the reduced dynamics on an invariant manifold
of the original system, we derive governing equations for the system which we
use to study stability of the incoherent states and the dynamical transitional
behavior from stable incoherent states to stable coherent states. We find that
spread in the distribution function of delays can greatly alter the system
dynamics. | nlin_CD |
Generalized analytical solutions and experimental confirmation of
complete synchronization in a class of mutually-coupled simple nonlinear
electronic circuits: In this paper, we present a novel explicit analytical solution for the
normalized state equations of mutually-coupled simple chaotic systems. A
generalized analytical solution is obtained for a class of simple nonlinear
electronic circuits with two different nonlinear elements. The synchronization
dynamics of the circuit systems were studied using the analytical solutions.
the analytical results thus obtained have been validated through numerical
simulation results. Further, we provide a sufficient condition for
synchronization in mutually-coupled, second-order simple chaotic systems
through an analysis on the eigenvalues of the difference system. The
bifurcation of the eigenvalues of the difference system as functions of the
coupling parameter in each of the piecewise-linear regions, revealing the
existence of stable synchronized states is presented. The stability of
synchronized states are studied using the {\emph{Master Stability Function}}.
Finally, the electronic circuit experimental results confirming the phenomenon
of complete synchronization in each of the circuit system is presented. | nlin_CD |
Spatial and Temporal Taylor's Law in 1-Dim Chaotic Maps: By using low-dimensional chaos maps, the power law relationship established
between the sample mean and variance called Taylor's Law (TL) is studied. In
particular, we aim to clarify the relationship between TL from the spatial
ensemble (STL) and the temporal ensemble (TTL). Since the spatial ensemble
corresponds to independent sampling from a stationary distribution, we confirm
that STL is explained by the skewness of the distribution. The difference
between TTL and STL is shown to be originated in the temporal correlation of a
dynamics. In case of logistic and tent maps, the quadratic relationship in the
mean and variance, called Bartlett's law, is found analytically. On the other
hand, TTL in the Hassell model can be well explained by the chunk structure of
the trajectory, whereas the TTL of the Ricker model have a different mechanism
originated from the specific form of the map. | nlin_CD |
Chaotic diffusion in the action and frequency domains: estimate of
instability times: Purpose: Chaotic diffusion in the non-linear systems is commonly studied in
the action framework. In this paper, we show that the study in the frequency
domain provides good estimates of the sizes of the chaotic regions in the phase
space, also as the diffusion timescales inside these regions.
Methods: Applying the traditional tools, such as Poincar\'e Surfaces of
Section, Lyapunov Exponents and Spectral Analysis, we characterise the phase
space of the Planar Circular Restricted Three Body Problem (PCR3BP). For the
purpose of comparison, the diffusion coefficients are obtained in the action
domain of the problem, applying the Shannon Entropy Method (SEM), also as in
the frequency domain, applying the Mean Squared Displacement (MSD) method and
Laskar's Equation of Diffusion. We compare the diffusion timescales defined by
the diffusion coefficients obtained to the Lyapunov times and the instability
times obtained through direct numerical integrations.
Results: Traditional tools for detecting chaos tend to misrepresent regimes
of motion, in which either slow-diffusion or confined-diffusion processes
dominates. The SEM shows a good performance in the regions of slow chaotic
diffusion, but it fails to characterise regions of strong chaotic motion. The
frequency-based methods are able to precisely characterise the whole phase
space and the diffusion times obtained in the frequency domain present
satisfactory agreement with direct integration instability times, both in weak
and strong chaotic motion regimes. The diffusion times obtained by means of the
SEM fail to match correctly the instability times provided by numerical
integrations.
Conclusion: We conclude that the study of dynamical instabilities in the
frequency domain provides reliable estimates of the diffusion timescales, and
also presents a good cost-benefit in terms of computation-time. | nlin_CD |
Hidden chaotic attractors in fractional-order systems: In this paper, we present a scheme for uncovering hidden chaotic attrac- tors
in nonlinear autonomous systems of fractional order. The stability of
equilibria of fractional-order systems is analyzed. The underlying initial
value problem is nu- merically integrated with the predictor-corrector
Adams-Bashforth-Moulton algo- rithm for fractional-order differential
equations. Three examples of fractional-order systems are considered: a
generalized Lorenz system, the Rabinovich-Fabrikant system and a non-smooth
Chua system. | nlin_CD |
Negative-coupling resonances in pump-coupled lasers: We consider coupled lasers, where the intensity deviations from the steady
state, modulate the pump of the other lasers. Most of our results are for two
lasers where the coupling constants are of opposite sign. This leads to a Hopf
bifurcation to periodic output for weak coupling. As the magnitude of the
coupling constants is increased (negatively) we observe novel amplitude effects
such as a weak coupling resonance peak and, strong coupling subharmonic
resonances and chaos. In the weak coupling regime the output is predicted by a
set of slow evolution amplitude equations. Pulsating solutions in the strong
coupling limit are described by discrete map derived from the original model. | nlin_CD |
Some aspects of the synchronization in coupled maps: Through numerical simulations we analyze the synchronization time and the
Lyapunov dimension of a coupled map lattice consisting of a chain of chaotic
logistic maps exhibiting power law interactions. From the observed behaviors we
find a lower bound for the size $N$ of the lattice, independent of the range
and strength of the interaction, which imposes a practical lower bound in
numerical simulations for the system to be considered in the thermodynamic
limit. We also observe the existence of a strong correlation between the
averaged synchronization time and the Lyapunov dimension. This is an
interesting result because it allows an analytical estimation of the
synchronization time, which otherwise requires numerical simulations. | nlin_CD |
Ray engineering from chaos to order in two-dimensional optical cavities: Chaos, namely exponential sensitivity to initial conditions, is generally
considered a nuisance, inasmuch as it prevents long-term predictions in
physical systems. Here, we present an easily accessible approach to undo
deterministic chaos and tailor ray trajectories in arbitrary two-dimensional
optical billiards, by introducing spatially varying refractive index therein. A
new refractive index landscape is obtained by a conformal mapping, which makes
the trajectories of the chaotic billiard fully predictable and the billiard
fully integrable. Moreover, trajectory rectification can be pushed a step
further by relating chaotic billiards with non-Euclidean geometries. Two
examples are illustrated by projecting billiards built on a sphere as well as
the deformed spacetime outside a Schwarzschild black hole, which respectively
lead to all periodic orbits and spiraling trajectories in the resulting 2D
billiards/cavities. An implementation of our method is proposed, which enables
real-time control of chaos and could further contribute to a wealth of
potential applications in the domain of optical microcavities. | nlin_CD |
A new spatio-temporal description of long-delayed systems: ruling the
dynamics: The data generated by long-delayed dynamical systems can be organized in
patterns by means of the so-called spatio-temporal representation, uncovering
the role of multiple time-scales as independent degrees of freedom. However,
their identification as equivalent space and time variables does not lead to a
correct dynamical rule. We introduce a new framework for a proper description
of the dynamics in the thermodynamic limit, providing a general avenue for the
treatment of long-delayed systems in terms of partial differential equations.
Such scheme is generic and does not depend on the vicinity of a super-critical
bifurcation as required in previous approaches. We discuss the general validity
and limit of this method and consider the exemplary cases of long-delayed
excitable, bistable and Landau systems. | nlin_CD |
Universal Velocity Profile for Coherent Vortices in Two-Dimensional
Turbulence: Two-dimensional turbulence generated in a finite box produces large-scale
coherent vortices coexisting with small-scale fluctuations. We present a
rigorous theory explaining the $\eta=1/4$ scaling in the $V\propto r^{-\eta}$
law of the velocity spatial profile within a vortex, where $r$ is the distance
from the vortex center. This scaling, consistent with earlier numerical and
laboratory measurements, is universal in its independence of details of the
small-scale injection of turbulent fluctuations and details of the shape of the
box. | nlin_CD |
Phase description of chaotic oscillators: This paper presents a phase description of chaotic dynamics for the study of
chaotic phase synchronization. A prominent feature of the proposed description
is that it systematically incorporates the dynamics of the non-phase variables
inherent in the system. Taking these non-phase dynamics into account is
essential for capturing the complicated nature of chaotic phase
synchronization, even in a qualitative manner. We numerically verified the
validity of the proposed description in application to the R\"{o}ssler and
Lorenz oscillators, and we found that our method provides an accurate
description of the characteristic distorted shapes of the synchronization
regions for these chaotic oscillators. Furthermore, the proposed description
allows us to systematically identify and describe the origin of this
distortion. | nlin_CD |
A PDE-Based Approach to Classical Phase-Space Deformations: This paper presents a PDE-based approach to finding an optimal canonical
basis with which to represent a nearly integrable Hamiltonian. The idea behind
the method is to continuously deform the initial canonical basis in such a way
that the dependence of the Hamiltonian on the canonical position of the final
basis is minimized. The final basis incorporates as much of the classical
dynamics as possible into an integrable Hamiltonian, leaving a much smaller
non-integrable component than in the initial representation. With this approach
it is also possible to construct the semiclassical wavefunctions corresponding
to the final canonical basis. This optimized basis is potentially useful in
quantum calculations, both as a way to minimize the required size of basis
sets, and as a way to provide physical insight by isolating those effects
resulting from integrable dynamics. | nlin_CD |
Discontinuous Attractor Dimension at the Synchronization Transition of
Time-Delayed Chaotic Systems: The attractor dimension at the transition to complete synchronization in a
network of chaotic units with time-delayed couplings is investigated. In
particular, we determine the Kaplan-Yorke dimension from the spectrum of
Lyapunov exponents for iterated maps and for two coupled semiconductor lasers.
We argue that the Kaplan-Yorke dimension must be discontinuous at the
transition and compare it to the correlation dimension. For a system of
Bernoulli maps we indeed find a jump in the correlation dimension. The
magnitude of the discontinuity in the Kaplan-Yorke dimension is calculated for
networks of Bernoulli units as a function of the network size. Furthermore the
scaling of the Kaplan-Yorke dimension as well as of the Kolmogorov entropy with
system size and time delay is investigated. | nlin_CD |
Anomalous exponents in the rapid-change model of the passive scalar
advection in the order $ε^{3}$: Field theoretic renormalization group is applied to the Kraichnan model of a
passive scalar advected by the Gaussian velocity field with the covariance
$<{\bf v}(t,{\bf x}){\bf v}(t',{\bf x})> - <{\bf v}(t,{\bf x}){\bf v}(t',{\bf
x'})> \propto\delta(t-t')|{\bf x}-{\bf x'} |^{\epsilon}$. Inertial-range
anomalous exponents, related to the scaling dimensions of tensor composite
operators built of the scalar gradients, are calculated to the order
$\epsilon^{3}$ of the $\epsilon$ expansion. The nature and the convergence of
the $\epsilon$ expansion in the models of turbulence is are briefly discussed. | nlin_CD |
Stability and bifurcations in an epidemic model with varying immunity
period: An epidemic model with distributed time delay is derived to describe the
dynamics of infectious diseases with varying immunity. It is shown that
solutions are always positive, and the model has at most two steady states:
disease-free and endemic. It is proved that the disease-free equilibrium is
locally and globally asymptotically stable. When an endemic equilibrium exists,
it is possible to analytically prove its local and global stability using
Lyapunov functionals. Bifurcation analysis is performed using DDE-BIFTOOL and
traceDDE to investigate different dynamical regimes in the model using
numerical continuation for different values of system parameters and different
integral kernels. | nlin_CD |
Intermittency, cascades and thin sets in three-dimensional Navier-Stokes
turbulence: Visual manifestations of intermittency in computations of three dimensional
Navier-Stokes fluid turbulence appear as the low-dimensional or `thin'
filamentary sets on which vorticity and strain accumulate as energy cascades
down to small scales. In order to study this phenomenon, the first task of this
paper is to investigate how weak solutions of the Navier-Stokes equations can
be associated with a cascade and, as a consequence, with an infinite sequence
of inverse length scales. It turns out that this sequence converges to a finite
limit. The second task is to show how these results scale with integer
dimension $D=1,\,2,\,3$ and, in the light of the occurrence of thin sets, to
discuss the mechanism of how the fluid might find the smoothest, most
dissipative class of solutions rather than the most singular. | nlin_CD |
Self-averaging characteristics of spectral fluctuations: The spectral form factor as well as the two-point correlator of the density
of (quasi-)energy levels of individual quantum dynamics are not self-averaging.
Only suitable smoothing turns them into useful characteristics of spectra. We
present numerical data for a fully chaotic kicked top, employing two types of
smoothing: one involves primitives of the spectral correlator, the second a
small imaginary part of the quasi-energy. Self-averaging universal (like the
CUE average) behavior is found for the smoothed correlator, apart from noise
which shrinks like $1\over\sqrt N$ as the dimension $N$ of the quantum Hilbert
space grows. There are periodically repeated quasi-energy windows of
correlation decay and revival wherein the smoothed correlation remains finite
as $N\to\infty$ such that the noise is negligible. In between those windows
(where the CUE averaged correlator takes on values of the order ${1\over N^2}$)
the noise becomes dominant and self-averaging is lost. We conclude that the
noise forbids distinction of CUE and GUE type behavior. Surprisingly, the
underlying smoothed generating function does not enjoy any self-averaging
outside the range of its variables relevant for determining the two-point
correlator (and certain higher-order ones). --- We corroborate our numerical
findings for the noise by analytically determining the CUE variance of the
smoothed single-matrix correlator. | nlin_CD |
Surrogate Test to Distinguish between Chaotic and Pseudoperiodic Time
Series: In this communication a new algorithm is proposed to produce surrogates for
pseudoperiodic time series. By imposing a few constraints on the noise
components of pseudoperiodic data sets, we devise an effective method to
generate surrogates. Unlike other algorithms, this method properly copes with
pseudoperiodic orbits contaminated with linear colored observational noise. We
will demonstrate the ability of this algorithm to distinguish chaotic orbits
from pseudoperiodic orbits through simulation data sets from theR\"{o}ssler
system. As an example of application of this algorithm, we will also employ it
to investigate a human electrocardiogram (ECG) record. | nlin_CD |
Capture into resonance and escape from it in a forced nonlinear pendulum: We study dynamics of a nonlinear pendulum under a periodic force with small
amplitude and slowly decreasing frequency. It is well known that when the
frequency of the external force passes through the value of the frequency of
the unperturbed pendulum's oscillations, the pendulum can be captured into the
resonance. The captured pendulum oscillates in such a way that the resonance is
preserved, and the amplitude of the oscillations accordingly grows. We consider
this problem in the frames of a standard Hamiltonian approach to resonant
phenomena in slow-fast Hamiltonian systems developed earlier, and evaluate the
probability of capture into the resonance. If the system passes the resonance
at small enough initial amplitudes of the pendulum, the capture occurs with
necessity (so-called autoresonance). In general, the probability of capture
varies between one and zero, depending on the initial amplitude. We demonstrate
that a pendulum captured at small values of its amplitude escapes from the
resonance in the domain of oscillations close to the separatrix of the
pendulum, and evaluate the amplitude of the oscillations at the escape. | nlin_CD |
Complex dynamical properties of coupled Van der Pol-Duffing oscillators
with balanced loss and gain: We consider a Hamiltonian system of coupled Van der Pol-Duffing(VdPD)
oscillators with balanced loss and gain. The system is analyzed perturbatively
by using Renormalization Group(RG) techniques as well as Multiple Scale
Analysis(MSA). Both the methods produce identical results in the leading order
of the perturbation. The RG flow equation is exactly solvable and the slow
variation of amplitudes and phases in time can be computed analytically. The
system is analyzed numerically and shown to admit periodic solutions in regions
of parameter-space, confirming the results of the linear stability analysis and
perturbation methods. The complex dynamical behavior of the system is studied
in detail by using time-series, Poincar$\acute{e}$-sections, power-spectra,
auto-correlation function and bifurcation diagrams. The Lyapunov exponents are
computed numerically. The numerical analysis reveals chaotic behaviour in the
system beyond a critical value of the parameter that couples the two VdPD
oscillators through linear coupling, thereby providing yet another example of
Hamiltonian chaos in a system with balanced loss and gain. Further, we modify
the nonlinear terms of the model to make it a non-Hamiltonian system of coupled
VdPD oscillators with balanced loss and gain. The non-Hamiltonian system is
analyzed perturbativly as well as numerically and shown to posses regular
periodic as well as chaotic solutions. It is seen that the
${\cal{PT}}$-symmetry is not an essential requirement for the existence of
regular periodic solutions in both the Hamiltonian as well as non-Hamiltonian
systems. | nlin_CD |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.