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From Turing patterns to chimera states in the 2D Brusselator model: The Brusselator has been used as a prototype model for autocatalytic
reactions, and in particular for the Belouzov- Zhabotinsky reaction. When
coupled at the diffusive limit, the Brusselator undergoes a Turing bifurcation
resulting in the formation of classical Turing patterns, such as spots, stripes
and spirals in 2 spatial dimensions. In the present study we use generic
nonlocally coupled Brusselators and show that in the limit of the coupling
range R->1 (diffusive limit), the classical Turing patterns are recovered,
while for intermediate coupling ranges and appropriate parameter values chimera
states are produced. This study demonstrates how the parameters of a typical
nonlinear oscillator can be tuned so that the coupled system passes from
spatially stable Turing structures to dynamical spatiotemporal chimera states. | nlin_CD |
A new approach to simulating stochastic delayed systems: In this paper we present a new method for deriving It\^{o} stochastic delay
differential equations (SDDEs) from delayed chemical master equations (DCMEs).
Considering alternative formulations of SDDEs that can be derived from the same
DCME, we prove that they are equivalent both in distribution, and in sample
paths they produce. This allows us to formulate an algorithmic approach to
deriving equivalent It\^{o} SDDEs with a smaller number of noise variables,
which increases the computational speed of simulating stochastic delayed
systems. The new method is illustrated on a simple model of two interacting
species, and it shows excellent agreement with the results of direct stochastic
simulations, while also demonstrating a much superior speed of performance. | nlin_CD |
Experimental and numerical investigation of the reflection coefficient
and the distributions of Wigner's reaction matrix for irregular graphs with
absorption: We present the results of experimental and numerical study of the
distribution of the reflection coefficient P(R) and the distributions of the
imaginary P(v) and the real P(u) parts of the Wigner's reaction K matrix for
irregular fully connected hexagon networks (graphs) in the presence of strong
absorption. In the experiment we used microwave networks, which were built of
coaxial cables and attenuators connected by joints. In the numerical
calculations experimental networks were described by quantum fully connected
hexagon graphs. The presence of absorption introduced by attenuators was
modelled by optical potentials. The distribution of the reflection coefficient
P(R) and the distributions of the reaction K matrix were obtained from the
measurements and numerical calculations of the scattering matrix S of the
networks and graphs, respectively. We show that the experimental and numerical
results are in good agreement with the exact analytic ones obtained within the
framework of random matrix theory (RMT). | nlin_CD |
A mechanical mode-stirred reverberation chamber with chaotic geometry: A previous research on multivariate approach to the calculation of
reverberation chamber correlation matrices is used to calculate the number of
independent positions in a mode-stirred reverberation chamber. Anomalies and
counterintuitive behavior are observed in terms of number of correlated matrix
elements with respect to increasing frequency. This is ascribed to the regular
geometry forming the baseline cavity (screened room) of a reverberation
chamber, responsible for localizing energy and preserving regular modes
(bouncing ball modes). Smooth wall deformations are introduced in order to
create underlying Lyapunov instability of rays and then destroy survived
regular modes. Numerical full-wave simulations are performed for a
reverberation chamber with corner hemispheres and (off-)center wall spherical
caps. Field sampling is performed by moving a mechanical carousel stirrer. It
is found that wave-chaos inspired baseline geometries improve chamber
performances in terms of lowest usable frequencies and number of independent
cavity realizations of mechanical stirrers. | nlin_CD |
Solution of linearized Fokker - Planck equation for incompressible fluid: In this work we construct algebraic equation for elements of spectrum of
linearized Fokker - Planck differential operator for incompressible fluid. We
calculate roots of this equation using simple numeric method. For all these
roots real part is positive, that is corresponding solutions are damping.
Eigenfunctions of linearized Fokker - Planck differential operator for
incompressible fluid are expressed as linear combinations of eigenfunctions of
usual Fokker - Planck differential operator. Poisson's equation for pressure is
derived from incompressibility condition. It is stated, that the pressure could
be totally eliminated from dynamics equations. The Cauchy problem setup and
solution method is presented. The role of zero pressure solutions as
eigenfunctions for confluent eigenvalues is emphasized. | nlin_CD |
Unstable periodic orbits in a chaotic meandering jet flow: We study the origin and bifurcations of typical classes of unstable periodic
orbits in a jet flow that was introduced before as a kinematic model of chaotic
advection, transport and mixing of passive scalars in meandering oceanic and
atmospheric currents. A method to detect and locate the unstable periodic
orbits and classify them by the origin and bifurcations is developed. We
consider in detail period-1 and period-4 orbits playing an important role in
chaotic advection. We introduce five classes of period-4 orbits: western and
eastern ballistic ones, whose origin is associated with ballistic resonances of
the fourth order, rotational ones, associated with rotational resonances of the
second and fourth orders, and rotational-ballistic ones associated with a
rotational-ballistic resonance. It is a new kind of nonlinear resonances that
may occur in chaotic flow with jets and/or circulation cells. Varying the
perturbation amplitude, we track out the origin and bifurcations of the orbits
for each class. | nlin_CD |
Random Matrix Spectra as a Time Series: Spectra of ordered eigenvalues of finite Random Matrices are interpreted as a
time series. Dataadaptive techniques from signal analysis are applied to
decompose the spectrum in clearly differentiated trend and fluctuation modes,
avoiding possible artifacts introduced by standard unfolding techniques. The
fluctuation modes are scale invariant and follow different power laws for
Poisson and Gaussian ensembles, which already during the unfolding allows to
distinguish the two cases. | nlin_CD |
Unified Model, and Novel Reverse Recovery Nonlinearities, of the Driven
Diode Resonator: We study the origins of period doubling and chaos in the driven series
resistor-inductor-varactor diode (RLD) nonlinear resonant circuit. We find that
resonators driven at frequencies much higher than the diode reverse recovery
rate do not show period doubling, and that models of chaos based on the
nonlinear capacitance of the varactor diode display a reverse-recovery-like
effect, and this effect strongly resembles reverse recovery of real diodes. We
find for the first time that in addition to the known dependence of the reverse
recovery time on past current maxima, there are also important nonlinear
dependencies on pulse frequency, duty-cycle, and DC voltage bias. Similar
nonlinearities are present in the nonlinear capacitance models of these diodes.
We conclude that a history-dependent and nonlinear reverse recovery time is an
essential ingredient for chaotic behavior of this circuit, and demonstrate for
the first time that all major competing models have this effect, either
explicitly or implicitly. Besides unifying the two major models of RLD chaos,
our work reveals that the nonlinearities of the reverse recovery time must be
included for a complete understanding of period doubling and chaos in this
circuit. | nlin_CD |
Arnold diffusion of charged particles in ABC magnetic fields: We prove the existence of diffusing solutions in the motion of a charged
particle in the presence of an ABC magnetic field. The equations of motion are
modeled by a 3DOF Hamiltonian system depending on two parameters. For small
values of these parameters, we obtain a normally hyperbolic invariant manifold
and we apply the so-called geometric methods for a priori unstable systems
developed by A. Delshams, R. de la Llave, and T.M. Seara. We characterize
explicitly sufficient conditions for the existence of a transition chain of
invariant tori having heteroclinic connections, thus obtaining global
instability (Arnold diffusion). We also check the obtained conditions in a
computer assisted proof. ABC magnetic fields are the simplest force-free type
solutions of the magnetohydrodynamics equations with periodic boundary
conditions, so our results are of potential interest in the study of the motion
of plasma charged particles in a tokamak. | nlin_CD |
Symmetry broken states in an ensemble of globally coupled pendulums: We consider the rotational dynamics in an ensemble of globally coupled
identical pendulums. This model is essentially a generalization of the standard
Kuramoto model, which takes into account the inertia and the intrinsic
nonlinearity of the community elements. There exists the wide variety of
in-phase and out-of-phase regimes. Many of these states appear due to broken
symmetry. In the case of small dissipation our theoretical analysis allows one
to find the boundaries of the instability domain of in-phase rotational mode
for ensembles with arbitrary number of pendulums, describe all arising
out-of-phase rotation modes and study in detail their stability. For the system
of three elements parameter sets corresponding to the unstable in-phase
rotations we find a number of out-of-phase regimes and investigate their
stability and bifurcations both analytically and numerically. As a result, we
obtain a sufficiently detailed picture of the symmetry breaking and existence
of various regular and chaotic states. | nlin_CD |
A moment-equation-copula-closure method for nonlinear vibrational
systems subjected to correlated noise: We develop a moment equation closure minimization method for the inexpensive
approximation of the steady state statistical structure of nonlinear systems
whose potential functions have bimodal shapes and which are subjected to
correlated excitations. Our approach relies on the derivation of moment
equations that describe the dynamics governing the two-time statistics. These
are combined with a non-Gaussian pdf representation for the joint
response-excitation statistics that has i) single time statistical structure
consistent with the analytical solutions of the Fokker-Planck equation, and ii)
two-time statistical structure with Gaussian characteristics. Through the
adopted pdf representation, we derive a closure scheme which we formulate in
terms of a consistency condition involving the second order statistics of the
response, the closure constraint. A similar condition, the dynamics constraint,
is also derived directly through the moment equations. These two constraints
are formulated as a low-dimensional minimization problem with respect to
unknown parameters of the representation, the minimization of which imposes an
interplay between the dynamics and the adopted closure. The new method allows
for the semi-analytical representation of the two-time, non-Gaussian structure
of the solution as well as the joint statistical structure of the
response-excitation over different time instants. We demonstrate its
effectiveness through the application on bistable nonlinear
single-degree-of-freedom energy harvesters with mechanical and electromagnetic
damping, and we show that the results compare favorably with direct Monte-Carlo
Simulations. | nlin_CD |
A taxonomy for generalized synchronization between flat-coupled systems: Generalized synchronization is plausibly the most complex form of
synchronization. Previous studies have revealed the existence of weak or strong
forms of generalized synchronization depending on the multi- or mono-valued
nature of the mapping between the attractors of two unidirectionally-coupled
systems. Generalized synchronization is here obtained by coupling two systems
with a flat control law. Here, we demonstrate that the corresponding
first-return maps can be topologically conjugate in some cases. Conversely, the
response map can foliated while the drive map is not. We describe the
corresponding types of generalized synchronization, explicitly focusing on the
influence of the coupling strength when significantly different dimensions or
dissipation properties characterize the coupled systems. A taxonomy of
generalized synchronization based on these properties is proposed. | nlin_CD |
Symbolic Synchronization and the Detection of Global Properties of
Coupled Dynamics from Local Information: We study coupled dynamics on networks using symbolic dynamics. The symbolic
dynamics is defined by dividing the state space into a small number of regions
(typically 2), and considering the relative frequencies of the transitions
between those regions. It turns out that the global qualitative properties of
the coupled dynamics can be classified into three different phases based on the
synchronization of the variables and the homogeneity of the symbolic dynamics.
Of particular interest is the {\it homogeneous unsynchronized phase} where the
coupled dynamics is in a chaotic unsynchronized state, but exhibits (almost)
identical symbolic dynamics at all the nodes in the network. We refer to this
dynamical behaviour as {\it symbolic synchronization}. In this phase, the local
symbolic dynamics of any arbitrarily selected node reflects global properties
of the coupled dynamics, such as qualitative behaviour of the largest Lyapunov
exponent and phase synchronization. This phase depends mainly on the network
architecture, and only to a smaller extent on the local chaotic dynamical
function. We present results for two model dynamics, iterations of the
one-dimensional logistic map and the two-dimensional H\'enon map, as local
dynamical function. | nlin_CD |
Chaotic dynamics with Maxima: We present an introduction to the study of chaos in discrete and continuous
dynamical systems using the CAS Maxima. These notes are intended to cover the
standard topics and techniques: discrete and continuous logistic equation to
model growth population, staircase plots, bifurcation diagrams and chaos
transition, nonlinear continuous dynamics (Lorentz system and Duffing
oscillator), Lyapunov exponents, Poincar\'e sections, fractal dimension and
strange attractors. The distinctive feature here is the use of free software
with just one ingredient: the CAS Maxima. It is cross-platform and have
extensive on-line documentation. | nlin_CD |
Accurately Estimating the State of a Geophysical System with Sparse
Observations: Predicting the Weather: Utilizing the information in observations of a complex system to make
accurate predictions through a quantitative model when observations are
completed at time $T$, requires an accurate estimate of the full state of the
model at time $T$.
When the number of measurements $L$ at each observation time within the
observation window is larger than a sufficient minimum value $L_s$, the
impediments in the estimation procedure are removed. As the number of available
observations is typically such that $L \ll L_s$, additional information from
the observations must be presented to the model.
We show how, using the time delays of the measurements at each observation
time, one can augment the information transferred from the data to the model,
removing the impediments to accurate estimation and permitting dependable
prediction. We do this in a core geophysical fluid dynamics model, the shallow
water equations, at the heart of numerical weather prediction. The method is
quite general, however, and can be utilized in the analysis of a broad spectrum
of complex systems where measurements are sparse. When the model of the complex
system has errors, the method still enables accurate estimation of the state of
the model and thus evaluation of the model errors in a manner separated from
uncertainties in the data assimilation procedure. | nlin_CD |
Chaotic Phenomenon in Nonlinear Gyrotropic Medium: Nonlinear gyrotropic medium is a medium, whose natural optical activity
depends on the intensity of the incident light wave. The Kuhn's model is used
to study nonlinear gyrotropic medium with great success. The Kuhn's model
presents itself a model of nonlinear coupled oscillators. This article is
devoted to the study of the Kuhn's nonlinear model. In the first paragraph of
the paper we study classical dynamics in case of weak as well as strong
nonlinearity. In case of week nonlinearity we have obtained the analytical
solutions, which are in good agreement with the numerical solutions. In case of
strong nonlinearity we have determined the values of those parameters for which
chaos is formed in the system under study. The second paragraph of the paper
refers to the question of the Kuhn's model integrability. It is shown, that at
the certain values of the interaction potential this model is exactly
integrable and under certain conditions it is reduced to so-called universal
Hamiltonian. The third paragraph of the paper is devoted to quantum-mechanical
consideration. It shows the possibility of stochastic absorption of external
field energy by nonlinear gyrotropic medium. The last forth paragraph of the
paper is devoted to generalization of the Kuhn's model for infinite chain of
interacting oscillators. | nlin_CD |
Theory of localization and resonance phenomena in the quantum kicked
rotor: We present an analytic theory of quantum interference and Anderson
localization in the quantum kicked rotor (QKR). The behavior of the system is
known to depend sensitively on the value of its effective Planck's constant
$\he$. We here show that for rational values of $\he/(4\pi)=p/q$, it bears
similarity to a disordered metallic ring of circumference $q$ and threaded by
an Aharonov-Bohm flux. Building on that correspondence, we obtain quantitative
results for the time--dependent behavior of the QKR kinetic energy, $E(\tilde
t)$ (this is an observable which sensitively probes the system's localization
properties). For values of $q$ smaller than the localization length $\xi$, we
obtain scaling $E(\tilde t) \sim \Delta \tilde t^2$, where $\Delta=2\pi/q$ is
the quasi--energy level spacing on the ring. This scaling is indicative of a
long time dynamics that is neither localized nor diffusive. For larger values
$q\gg \xi$, the functions $E(\tilde t)\to \xi^2$ saturates (up to exponentially
small corrections $\sim\exp(-q/\xi)$), thus reflecting essentially localized
behavior. | nlin_CD |
Chaos synchronization with coexisting global fields: We investigate the phenomenon of chaos synchronization in systems subject to
coexisting autonomous and external global fields by employing a simple model of
coupled maps. Two states of chaos synchronization are found: (i) complete
synchronization, where the maps synchronize among themselves and to the
external field, and (ii) generalized or internal synchronization, where the
maps synchronize among themselves but not to the external global field. We show
that the stability conditions for both states can be achieved for a system of
minimum size of two maps. We consider local maps possessing robust chaos and
characterize the synchronization states on the space of parameters of the
system. The state of generalized synchronization of chaos arises even the drive
and the local maps have the same functional form. This behavior is similar to
the process of spontaneous ordering against an external field found in
nonequilibrium systems. | nlin_CD |
Dynamics of rolling disk: In the paper we present the qualitative analysis of rolling motion without
slipping of a homogeneous round disk on a horisontal plane. The problem was
studied by S.A. Chaplygin, P. Appel and D. Korteweg who showed its
integrability. The behavior of the point of contact on a plane is investigated
and conditions under which its trajectory is finit are obtained. The
bifurcation diagrams are constructed. | nlin_CD |
Low dimensional behavior in three-dimensional coupled map lattices: The analysis of one-, two-, and three-dimensional coupled map lattices is
here developed under a statistical and dynamical perspective. We show that the
three-dimensional CML exhibits low dimensional behavior with long range
correlation and the power spectrum follows $1/f$ noise. This approach leads to
an integrated understanding of the most important properties of these universal
models of spatiotemporal chaos. We perform a complete time series analysis of
the model and investigate the dependence of the signal properties by change of
dimension. | nlin_CD |
Duffing-type equations: singular points of amplitude profiles and
bifurcations: We study the Duffing equation and its generalizations with polynomial
nonlinearities. Recently, we have demonstrated that metamorphoses of the
amplitude response curves, computed by asymptotic methods in implicit form as
$F\left( \Omega ,\ A\right) =0$, permit prediction of qualitative changes of
dynamics occurring at singular points of the implicit curve $F\left(\Omega ,\
A\right) =0$. In the present work we determine a global structure of singular
points of the amplitude profiles computing bifurcation sets, i.e. sets
containing all points in the parameter space for which the amplitude profile
has a singular point. We connect our work with independent research on
tangential points on amplitude profiles, associated with jump phenomena,
characteristic for the Duffing equation. We also show that our techniques can
be applied to solutions of form $\Omega _{\pm }=f_{\pm }\left( A\right) $,
obtained within other asymptotic approaches. | nlin_CD |
Characteristic times for the Fermi-Ulam Model: The mean Poincarr\'e recurrence time as well as the Lyapunov time are
measured for the Fermi-Ulam model. We confirm the mean recurrence time is
dependent on the size of the window chosen in the phase space to where
particles are allowed to recur. The fractal dimension of the region is
determined by the slope of the recurrence time against the size of the window
and two numerical values were measured: (i) $\mu$ = 1 confirming normal
diffusion for chaotic regions far from periodic domains and; (ii) $\mu$ = 2
leading to anomalous diffusion measured near periodic regions, a signature of
local trapping of an ensemble of particles. The Lyapunov time is measured over
different domains in the phase space through a direct determination of the
Lyapunov exponent, indeed being defined as its inverse. | nlin_CD |
Generalized Chaotic Synchronizationin Coupled Ginzburg-Landau Equations: Generalized synchronization is analyzed in unidirectionally coupled
oscillatory systems exhibiting spatiotemporal chaotic behavior described by
Ginzburg-Landau equations. Several types of coupling betweenthe systems are
analyzed. The largest spatial Lyapunov exponent is proposed as a new
characteristic of the state of a distributed system, and its calculation is
described for a distributed oscillatory system. Partial generalized
synchronization is introduced as a new type of chaotic synchronization in
spatially nonuniform distributed systems. The physical mechanisms responsible
for the onset of generalized chaotic synchronization in spatially distributed
oscillatory systems are elucidated. It is shown that the onset of generalized
chaotic synchronization is described by a modified Ginzburg-Landau equation
with additional dissipation irrespective of the type of coupling. The effect of
noise on the onset of a generalized synchronization regime in coupled
distributed systems is analyzed. | nlin_CD |
Naimark-Sacker Bifurcations in Linearly Coupled Quadratic Maps: We report exact analytical expressions locating the $0\to1$, $1\to2$ and
$2\to4$ bifurcation curves for a prototypical system of two linearly coupled
quadratic maps. Of interest is the precise location of the parameter sets where
Naimark-Sacker bifurcations occur, starting from a non-diagonal period-2 orbit.
This result is the key to understand the onset of synchronization in networks
of quadratic maps. | nlin_CD |
Scaling Analysis and Evolution Equation of the North Atlantic
Oscillation Index Fluctuations: The North Atlantic Oscillation (NAO) monthly index is studied from 1825 till
2002 in order to identify the scaling ranges of its fluctuations upon different
delay times and to find out whether or not it can be regarded as a Markov
process. A Hurst rescaled range analysis and a detrended fluctuation analysis
both indicate the existence of weakly persistent long range time correlations
for the whole scaling range and time span hereby studied. Such correlations are
similar to Brownian fluctuations. The Fokker-Planck equation is derived and
Kramers-Moyal coefficients estimated from the data. They are interpreted in
terms of a drift and a diffusion coefficient as in fluid mechanics. All partial
distribution functions of the NAO monthly index fluctuations have a form close
to a Gaussian, for all time lags, in agreement with the findings of the scaling
analyses. This indicates the lack of predictive power of the present NAO
monthly index. Yet there are some deviations for large (and thus rare) events.
Whence suggestions for other measurements are made if some improved
predictability of the weather/climate in the North Atlantic is of interest. The
subsequent Langevin equation of the NAO signal fluctuations is explicitly
written in terms of the diffusion and drift parameters, and a characteristic
time scale for these is given in appendix. | nlin_CD |
Exact geometric theory of dendronized polymer dynamics: Dendronized polymers consist of an elastic backbone with a set of iterated
branch structures (dendrimers)attached at every base point of the backbone. The
conformations of such molecules depend on the elastic deformation of the
backbone and the branches, as well as on nonlocal (e.g., electrostatic, or
Lennard-Jones) interactions between the elementary molecular units comprising
the dendrimers and/or backbone. We develop a geometrically exact theory for the
dynamics of such polymers, taking into account both local (elastic) and
nonlocal interactions. The theory is based on applying symmetry reduction of
Hamilton's principle for a Lagrangian defined on the tangent bundle of iterated
semidirect products of the rotation groups that represent the relative
orientations of the dendritic branches of the polymer. The resulting
symmetry-reduced equations of motion are written in conservative form. | nlin_CD |
Rich dynamics and anticontrol of extinction in a prey-predator system: This paper reveals some new and rich dynamics of a two-dimensional
prey-predator system and to anticontrol the extinction of one of the species.
For a particular value of the bifurcation parameter, one of the system variable
dynamics is going to extinct, while another remains chaotic. To prevent the
extinction, a simple anticontrol algorithm is applied so that the system orbits
can escape from the vanishing trap. As the bifurcation parameter increases, the
system presents quasiperiodic, stable, chaotic and also hyperchaotic orbits.
Some of the chaotic attractors are Kaplan-Yorke type, in the sense that the sum
of its Lyapunov exponents is positive. Also, atypically for undriven discrete
systems, it is numerically found that, for some small parameter ranges, the
system seemingly presents strange nonchaotic attractors. It is shown both
analytically and by numerical simulations that the original system and the
anticontrolled system undergo several Neimark-Sacker bifurcations. Beside the
classical numerical tools for analyzing chaotic systems, such as phase
portraits, time series and power spectral density, the 0-1 test is used to
differentiate regular attractors from chaotic attractors. | nlin_CD |
The bifurcations of the critical points and the role of the depth in a
symmetric Caldera potential energy surface: In this work, we continue the study of the bifurcations of the critical
points in a symmetric Caldera potential energy surface. In particular, we study
the influence of the depth of the potential on the trajectory behavior before
and after the bifurcations of the critical points. We observe two different
types of trajectory behavior: dynamical matching and the non-existence of
dynamical matching. Dynamical matching is a phenomenon that limits the way in
which a trajectory can exit the Caldera based solely on how it enters the
Caldera. Furthermore, we discuss two different types of symmetric Caldera
potential energy surface and the transition from the one type to the other
through the bifurcations of the critical points. | nlin_CD |
Chaotic motion of charged particles in toroidal magnetic configurations: We study the motion of a charged particle in a tokamak magnetic field and
discuss its chaotic nature. Contrary to most of recent studies, we do not make
any assumption on any constant of the motion and solve numerically the
cyclotron gyration using Hamiltonian formalism. We take advantage of a
symplectic integrator allowing us to make long-time simulations. First
considering an idealized magnetic configuration, we add a non generic
perturbation corresponding to a magnetic ripple, breaking one of the invariant
of the motion. Chaotic motion is then observed and opens questions about the
link between chaos of magnetic field lines and chaos of particle trajectories.
Second, we return to a axi-symmetric configuration and tune the safety factor
(magnetic configuration) in order to recover chaotic motion. In this last
setting with two constants of the motion, the presence of chaos implies that no
third global constant exists, we highlight this fact by looking at variations
of the first order of the magnetic moment in this chaotic setting. We are
facing a mixed phase space with both regular and chaotic regions and point out
the difficulties in performing a global reduction such as gyrokinetics. | nlin_CD |
Logarithmic periodicities in the bifurcations of type-I intermittent
chaos: The critical relations for statistical properties on saddle-node bifurcations
are shown to display undulating fine structure, in addition to their known
smooth dependence on the control parameter. A piecewise linear map with the
type-I intermittency is studied and a log-periodic dependence is numerically
obtained for the average time between laminar events, the Lyapunov exponent and
attractor moments. The origin of the oscillations is built in the natural
probabilistic measure of the map and can be traced back to the existence of
logarithmically distributed discrete values of the control parameter giving
Markov partition. Reinjection and noise effect dependences are discussed and
indications are given on how the oscillations are potentially applicable to
complement predictions made with the usual critical exponents, taken from data
in critical phenomena. | nlin_CD |
Estimation of initial conditions from a scalar time series: We introduce a method to estimate the initial conditions of a mutivariable
dynamical system from a scalar signal. The method is based on a modified
multidimensional Newton-Raphson method which includes the time evolution of the
system. The method can estimate initial conditions of periodic and chaotic
systems and the required length of scalar signal is very small. Also, the
method works even when the conditional Lyapunov exponent is positive. An
important application of our method is that synchronization of two chaotic
signals using a scalar signal becomes trivial and instantaneous. | nlin_CD |
Pseudo resonance induced quasi-periodic behavior in stochastic threshold
dynamics: Here we present a simple stochastic threshold model consisting of a
deterministic slowly decaying term and a fast stochastic noise term. The
process shows a pseudo-resonance, in the sense that for small and large
intensities of the noise the signal is irregular and the distribution of
threshold crossings is broad, while for a tuned intermediate value of noise
intensity the signal becomes quasi-periodic and the distribution of threshold
crossings is narrow. The mechanism captured by the model might be relevant for
explaining apparent quasi-periodicity of observed climatic variations where no
internal or external periodicities can be identified. | nlin_CD |
Universality and Hysteresis in Slow Sweeping of Bifurcations: Bifurcations in dynamical systems are often studied experimentally and
numerically using a slow parameter sweep. Focusing on the cases of
period-doubling and pitchfork bifurcations in maps, we show that the adiabatic
approximation always breaks down sufficiently close to the bifurcation, so that
the upsweep and downsweep dynamics diverge from one another, disobeying
standard bifurcation theory. Nevertheless, we demonstrate universal upsweep and
downsweep trajectories for sufficiently slow sweep rates, revealing that the
slow trajectories depend essentially on a structural asymmetry parameter, whose
effect is negligible for the stationary dynamics. We obtain explicit asymptotic
expressions for the universal trajectories, and use them to calculate the area
of the hysteresis loop enclosed between the upsweep and downsweep trajectories
as a function of the asymmetry parameter and the sweep rate. | nlin_CD |
Sampling Chaotic Trajectories Quickly in Parallel: The parallel computational complexity of the quadratic map is studied. A
parallel algorithm is described that generates typical pseudotrajectories of
length t in a time that scales as log t and increases slowly in the accuracy
demanded of the pseudotrajectory. Long pseudotrajectories are created in
parallel by putting together many short pseudotrajectories; Monte Carlo
procedures are used to eliminate the discontinuities between these short
pseudotrajectories and then suitably randomize the resulting long
pseudotrajectory. Numerical simulations are presented that show the scaling
properties of the parallel algorithm. The existence of the fast parallel
algorithm provides a way to formalize the intuitive notion that chaotic systems
do not generate complex histories. | nlin_CD |
Hiding message in Delay Time: Encryption with Synchronized time-delayed
systems: We propose a new communication scheme that uses time-delayed chaotic systems
with delay time modulation. In this method, the transmitter encodes a message
as an additional modulation of the delay timeand then the receiver decodes the
message by tracking the delay time.We demonstrate our communication scheme in a
system of coupled logistic maps.Also we discuss the error of the transferred
message due to an external noiseand present its correction method. | nlin_CD |
Non-Reversible Evolution of Quantum Chaotic System. Kinetic Description: Time dependent dynamics of the chaotic quantum-mechanical system has been
studied. Irreversibility of the dynamics is shown. It is shown, that being in
the initial moment in pure quantum-mechanical state, system makes irreversible
transition into mixed state. Original mechanism of mixed state formation is
offered. Quantum kinetic equation is obtained. Growth of the entropy during the
evolution process is estimated. | nlin_CD |
Classification and stability of simple homoclinic cycles in R^5: The paper presents a complete study of simple homoclinic cycles in R^5. We
find all symmetry groups Gamma such that a Gamma-equivariant dynamical system
in R^5 can possess a simple homoclinic cycle. We introduce a classification of
simple homoclinic cycles in R^n based on the action of the system symmetry
group. For systems in R^5, we list all classes of simple homoclinic cycles. For
each class, we derive necessary and sufficient conditions for asymptotic
stability and fragmentary asymptotic stability in terms of eigenvalues of
linearisation near the steady state involved in the cycle. For any action of
the groups Gamma which can give rise to a simple homoclinic cycle, we list
classes to which the respective homoclinic cycles belong, thus determining
conditions for asymptotic stability of these cycles. | nlin_CD |
Determination of fractal dimensions of solar radio bursts: We present a dimension analysis of a set of solar type I storms and type IV
events with different kind of fine structures, recorded at the Trieste
Astronomical Observatory. The signature of such types of solar radio events is
highly structured in time. However, periodicities are rather seldom, and linear
mode theory can provide only limited interpretation of the data. Therefore, we
performed an analysis based on methods of the nonlinear dynamics theory.
Additionally to the commonly used correlation dimension, we also calculated
local pointwise dimensions. This alternative approach is motivated by the fact
that astrophysical time series represent real-world systems, which cannot be
kept in a controlled state and which are highly interconnected with their
surroundings. In such systems pure determinism is rather unlikely to be
realized, and therefore a characterization by invariants of the dynamics might
probably be inadequate. In fact, the outcome of the dimension analysis does not
give hints for low-dimensional determinism in the data, but we show that,
contrary to the correlation dimension method, local dimension estimations can
give physical insight into the events even in cases in which pure determinism
cannot be established. In particular, in most of the analyzed radio events
nonlinearity in the data is detected, and the local dimension analysis provides
a basis for a quantitative description of the time series, which can be used to
characterize the complexity of the related physical system in a comparative and
non-invariant manner. | nlin_CD |
Bubbling in delay-coupled lasers: We theoretically study chaos synchronization of two lasers which are
delay-coupled via an active or a passive relay. While the lasers are
synchronized, their dynamics is identical to a single laser with delayed
feedback for a passive relay and identical to two delay-coupled lasers for an
active relay. Depending on the coupling parameters the system exhibits
bubbling, i.e., noise-induced desynchronization, or on-off intermittency. We
associate the desynchronization dynamics in the coherence collapse and low
frequency fluctuation regimes with the transverse instability of some of the
compound cavity's antimodes. Finally, we demonstrate how, by using an active
relay, bubbling can be suppressed. | nlin_CD |
Synchronization transitions in globally coupled rotors in presence of
noise and inertia: Exact results: We study a generic model of globally coupled rotors that includes the effects
of noise, phase shift in the coupling, and distributions of moments of inertia
and natural frequencies of oscillation. As particular cases, the setup includes
previously studied Sakaguchi-Kuramoto, Hamiltonian and Brownian mean-field, and
Tanaka-Lichtenberg-Oishi and Acebr\'on-Bonilla-Spigler models. We derive an
exact solution of the self-consistent equations for the order parameter in the
stationary state, valid for arbitrary parameters in the dynamics, and
demonstrate nontrivial phase transitions to synchrony that include reentrant
synchronous regimes. | nlin_CD |
Optimal Tree for Both Synchronizability and Converging Time: It has been proved that the spanning tree from a given network has the
optimal synchronizability, which means the index $R=\lambda_{N}/\lambda_{2}$
reaches the minimum 1. Although the optimal synchronizability is corresponding
to the minimal critical overall coupling strength to reach synchronization, it
does not guarantee a shorter converging time from disorder initial
configuration to synchronized state. In this letter, we find that it is the
depth of the tree that affects the converging time. In addition, we present a
simple and universal way to get such an effective oriented tree in a given
network to reduce the converging time significantly by minimizing the depth of
the tree. The shortest spanning tree has both the maximal synchronizability and
efficiency. | nlin_CD |
Reply to a Comment by J. Bolte, R. Glaser and S. Keppeler on:
Semiclassical theory of spin-orbit interactions using spin coherent states: We reply to a Comment on our recently proposed semiclassical theory for
systems with spin-orbit interactions. | nlin_CD |
Quantum fluctuations stabilize an inverted pendulum: We explore analytically the quantum dynamics of a point mass pendulum using
the Heisenberg equation of motion. Choosing as variables the mean position of
the pendulum, a suitably defined generalised variance and a generalised
skewness, we set up a dynamical system which reproduces the correct limits of
simple harmonic oscillator like and free rotor like behaviour. We then find the
unexpected result that the quantum pendulum released from and near the inverted
position executes oscillatory motion around the classically unstable position
provided the initial wave packet has a variance much greater than the variance
of the well known coherent state of the simple harmonic oscillator. The
behaviour of the dynamical system for the quantum pendulum is a higher
dimensional analogue of the behaviour of the Kapitza pendulum where the point
of support is vibrated vertically with a frequency higher than the critical
value needed to stabilize the inverted position. A somewhat similar phenomenon
has recently been observed in the non equilibrium dynamics of a spin - 1
Bose-Einstein Condensate. | nlin_CD |
A Universal Map for Fractal Structures in Weak Solitary Wave
Interactions: Fractal scatterings in weak solitary wave interactions is analyzed for
generalized nonlinear Schr\"odiger equations (GNLS). Using asymptotic methods,
these weak interactions are reduced to a universal second-order map. This map
gives the same fractal scattering patterns as those in the GNLS equations both
qualitatively and quantitatively. Scaling laws of these fractals are also
derived. | nlin_CD |
Desynchronization of systems of coupled Hindmarsh-Rose oscillators: It is widely assumed that neural activity related to synchronous rhythms of
large portions of neurons in specific locations of the brain is responsible for
the pathology manifested in patients' uncontrolled tremor and other similar
diseases. To model such systems Hindmarsh-Rose (HR) oscillators are considered
as appropriate as they mimic the qualitative behaviour of neuronal firing. Here
we consider a large number of identical HR-oscillators interacting through the
mean field created by the corresponding components of all oscillators.
Introducing additional coupling by feedback of Pyragas type, proportional to
the difference between the current value of the mean-field and its value some
time in the past, Rosenblum and Pikovsky (Phys. Rev. E 70, 041904, 2004)
demonstrated that the desirable desynchronization could be achieved with
appropriate set of parameters for the system. Following our experience with
stabilization of unstable steady states in dynamical systems, we show that by
introducing a variable delay, desynchronization is obtainable for much wider
range of parameters and that at the same time it becomes more pronounced. | nlin_CD |
On the inadequacy of the logistic map for cryptographic applications: This paper analyzes the use of the logistic map for cryptographic
applications. The most important characteristics of the logistic map are shown
in order to prove the inconvenience of considering this map in the design of
new chaotic cryptosystems. | nlin_CD |
Collection of Master-Slave Synchronized Chaotic Systems: In this work the open-plus-closed-loop (OPCL) method of synchronization is
used in order to synchronize the systems from the Sprott's collection of the
simplest chaotic systems. The method is general and we were looking for the
simplest coupling between master and slave system. The interval of parameters
were synchronization is achieved are obtained analytically using Routh-Hurwitz
conditions. Detailed calculations and numerical simulation are given for the
system I from the Sprott's collection. Working in the same manner for
non-linear systems based on ordinary differential equations the method can be
adopted for the teaching of the topic. | nlin_CD |
Drastic facilitation of the onset of global chaos in a periodically
driven Hamiltonian system due to an extremum in the dependence of
eigenfrequency on energy: The Chirikov resonance-overlap criterion predicts the onset of global chaos
if nonlinear resonances overlap in energy, which is conventionally assumed to
require a non-small magnitude of perturbation. We show that, for a
time-periodic perturbation, the onset of global chaos may occur at unusually
{\it small} magnitudes of perturbation if the unperturbed system possesses more
than one separatrix. The relevant scenario is the combination of the overlap in
the phase space between resonances of the same order and their overlap in
energy with chaotic layers associated with separatrices of the unperturbed
system. One of the most important manifestations of this effect is a drastic
increase of the energy range involved into the unbounded chaotic transport in
spatially periodic system driven by a rather {\it weak} time-periodic force,
provided the driving frequency approaches the extremal eigenfrequency or its
harmonics. We develop the asymptotic theory and verify it in simulations. | nlin_CD |
On the detuned 2:4 resonance: We consider families of Hamiltonian systems in two degrees of freedom with an
equilibrium in 1:2 resonance. Under detuning, this "Fermi resonance" typically
leads to normal modes losing their stability through period-doubling
bifurcations. For cubic potentials this concerns the short axial orbits and in
galactic dynamics the resulting stable periodic orbits are called "banana"
orbits. Galactic potentials are symmetric with respect to the co-ordinate
planes whence the potential -- and the normal form -- both have no cubic terms.
This $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetry turns the 1:2 resonance into a
higher order resonance and one therefore also speaks of the 2:4 resonance. In
this paper we study the 2:4 resonance in its own right, not restricted to
natural Hamiltonian systems where $H = T + V$ would consist of kinetic and
(positional) potential energy. The short axial orbit then turns out to be
dynamically stable everywhere except at a simultaneous bifurcation of banana
and "anti-banana" orbits, while it is now the long axial orbit that loses and
regains stability through two successive period-doubling bifurcations. | nlin_CD |
Anomalous correlators, "ghost" waves and nonlinear standing waves in the
$β$-FPUT system: We show that Hamiltonian nonlinear dispersive wave systems with cubic
nonlinearity and random initial data develop, during their evolution, anomalous
correlators. These are responsible for the appearance of "ghost" excitations,
i.e. those characterized by negative frequencies, in addition to the positive
ones predicted by the linear dispersion relation. We use generalization of the
Wick's decomposition and the wave turbulence theory to explain theoretically
the existence of anomalous correlators. We test our theory on the celebrated
$\beta$-Fermi-Pasta-Ulam-Tsingou chain and show that numerically measured
values of the anomalous correlators agree, in the weakly nonlinear regime, with
our analytical predictions. We also predict that similar phenomena will occur
in other nonlinear systems dominated by nonlinear interactions, including
surface gravity waves. Our results pave the road to study phase correlations in
the Fourier space for weakly nonlinear dispersive wave systems. | nlin_CD |
A new model of variable-length coupled pendulums: from hyperchaos to
superintegrability: This paper studies the dynamics and integrability of a variable-length
coupled pendulum system. The complexity of the model is presented by joining
various numerical methods, such as the Poincar\'e cross-sections,
phase-parametric diagrams, and Lyapunov exponents spectra. We show that the
presented model is hyperchaotic, which ensures its nonintegrability. We gave
analytical proof of this fact analyzing properties of the differential Galois
group of variational equations along certain particular solutions of the
system. We employ the Kovacic algorithm and its extension to dimension four to
analyze the differential Galois group. Amazingly enough, in the absence of the
gravitational potential and for certain values of the parameters, the system
can exhibit chaotic, integrable, as well as superintegrable dynamics. To the
best of our knowledge, this is the first attempt to use the method of Lyapunov
exponents in the systematic search for the first integrals of the system. We
show how to effectively apply the Lyapunov exponents as an indicator of
integrable dynamics. The explicit forms of integrable and superintegrable
systems are given.
The article has been published in Nonlinear Dynamics, and the final version
is available at this link: https://doi.org/10.1007/s11071-023-09253-5 | nlin_CD |
Cycle expansions for intermittent maps: In a generic dynamical system chaos and regular motion coexist side by side,
in different parts of the phase space. The border between these, where
trajectories are neither unstable nor stable but of marginal stability,
manifests itself through intermittency, dynamics where long periods of nearly
regular motions are interrupted by irregular chaotic bursts. We discuss the
Perron-Frobenius operator formalism for such systems, and show by means of a
1-dimensional intermittent map that intermittency induces branch cuts in
dynamical zeta functions. Marginality leads to long-time dynamical
correlations, in contrast to the exponentially fast decorrelations of purely
chaotic dynamics. We apply the periodic orbit theory to quantitative
characterization of the associated power-law decays. | nlin_CD |
Recovery time after localized perturbations in complex dynamical
networks: Maintaining the synchronous motion of dynamical systems interacting on
complex networks is often critical to their functionality. However, real-world
networked dynamical systems operating synchronously are prone to random
perturbations driving the system to arbitrary states within the corresponding
basin of attraction, thereby leading to epochs of desynchronized dynamics with
a priori unknown durations. Thus, it is highly relevant to have an estimate of
the duration of such transient phases before the system returns to synchrony,
following a random perturbation to the dynamical state of any particular node
of the network. We address this issue here by proposing the framework of
\emph{single-node recovery time} (SNRT) which provides an estimate of the
relative time scales underlying the transient dynamics of the nodes of a
network during its restoration to synchrony. We utilize this in differentiating
the particularly \emph{slow} nodes of the network from the relatively
\emph{fast} nodes, thus identifying the critical nodes which when perturbed
lead to significantly enlarged recovery time of the system before resuming
synchronized operation. Further, we reveal explicit relationships between the
SNRT values of a network, and its \emph{global relaxation time} when starting
all the nodes from random initial conditions. We employ the proposed concept
for deducing microscopic relationships between topological features of nodes
and their respective SNRT values. The framework of SNRT is further extended to
a measure of resilience of the different nodes of a networked dynamical system.
We demonstrate the potential of SNRT in networks of R\"{o}ssler oscillators on
paradigmatic topologies and a model of the power grid of the United Kingdom
with second-order Kuramoto-type nodal dynamics illustrating the conceivable
practical applicability of the proposed concept. | nlin_CD |
On asymptotic properties of some complex Lorenz-like systems: The classical Lorenz lowest order system of three nonlinear ordinary
differential equations, capable of producing chaotic solutions, has been
generalized by various authors in two main directions: (i) for number of
equations larger than three (Curry1978) and (ii) for the case of complex
variables and parameters. Problems of laser physics and geophysical fluid
dynamics (baroclinic instability, geodynamic theory, etc. - see the references)
can be related to this second aspect of generalization. In this paper we study
the asymptotic properties of some complex Lorenz systems, keeping in the mind
the physical basis of the model mathematical equations. | nlin_CD |
Sensitivity Analysis of Separation Time Along Weak Stability Boundary
Transfers: This study analyzes the sensitivity of the dynamics around Weak Stability
Boundary Transfers (WSBT) in the elliptical restricted three-body problem. With
WSBTs increasing popularity for cislunar transfers, understanding its
inherently chaotic dynamics becomes pivotal for guiding and navigating
cooperative spacecrafts as well as detecting non-cooperative objects. We
introduce the notion of separation time to gauge the deviation of a point near
a nominal WSBT from the trajectory's vicinity. Employing the Cauchy-Green
tensor to identify stretching directions in position and velocity, the
separation time, along with the Finite-Time Lyapunov Exponent are studied
within a ball of state uncertainty scaled to typical orbit determination
performances. | nlin_CD |
Turbulent boundary layer equations: We study a boundary layer problem for the Navier-Stokes-alpha model obtaining
a generalization of the Prandtl equations conjectured to represent the averaged
flow in a turbulent boundary layer. We solve the equations for the
semi-infinite plate, both theoretically and numerically. The latter solutions
agree with some experimental data in the turbulent boundary layer. | nlin_CD |
On universality of algebraic decays in Hamiltonian systems: Hamiltonian systems with a mixed phase space typically exhibit an algebraic
decay of correlations and of Poincare' recurrences, with numerical experiments
over finite times showing system-dependent power-law exponents. We conjecture
the existence of a universal asymptotic decay based on results for a Markov
tree model with random scaling factors for the transition probabilities.
Numerical simulations for different Hamiltonian systems support this conjecture
and permit the determination of the universal exponent. | nlin_CD |
Amplitude death in a ring of nonidentical nonlinear oscillators with
unidirectional coupling: We study the collective behaviors in a ring of coupled nonidentical nonlinear
oscillators with unidirectional coupling, of which natural frequencies are
distributed in a random way. We find the amplitude death phenomena in the case
of unidirectional couplings and discuss the differences between the cases of
bidirectional and unidirectional couplings. There are three main differences;
there exists neither partial amplitude death nor local clustering behavior but
oblique line structure which represents directional signal flow on the
spatio-temporal patterns in the unidirectional coupling case. The
unidirectional coupling has the advantage of easily obtaining global amplitude
death in a ring of coupled oscillators with randomly distributed natural
frequency. Finally, we explain the results using the eigenvalue analysis of
Jacobian matrix at the origin and also discuss the transition of dynamical
behavior coming from connection structure as coupling strength increases. | nlin_CD |
Cryptanalysis of a chaotic block cipher with external key and its
improved version: Recently, Pareek et al. proposed a symmetric key block cipher using multiple
one-dimensional chaotic maps. This paper reports some new findings on the
security problems of this kind of chaotic cipher: 1) a number of weak keys
exists; 2) some important intermediate data of the cipher are not sufficiently
random; 3) the whole secret key can be broken by a known-plaintext attack with
only 120 consecutive known plain-bytes in one known plaintext. In addition, it
is pointed out that an improved version of the chaotic cipher proposed by Wei
et al. still suffers from all the same security defects. | nlin_CD |
Lagrangian transport through surfaces in volume-preserving flows: Advective transport of scalar quantities through surfaces is of fundamental
importance in many scientific applications. From the Eulerian perspective of
the surface it can be quantified by the well-known integral of the flux
density. The recent development of highly accurate semi-Lagrangian methods for
solving scalar conservation laws and of Lagrangian approaches to coherent
structures in turbulent (geophysical) fluid flows necessitate a new approach to
transport from the (Lagrangian) material perspective. We present a Lagrangian
framework for calculating transport of conserved quantities through a given
surface in $n$-dimensional, fully aperiodic, volume-preserving flows. Our
approach does not involve any dynamical assumptions on the surface or its
boundary. | nlin_CD |
Big Entropy Fluctuations in Nonequilibrium Steady State: A Simple Model
with Gauss Heat Bath: Large entropy fluctuations in a nonequilibrium steady state of classical
mechanics were studied in extensive numerical experiments on a simple 2-freedom
model with the so-called Gauss time-reversible thermostat. The local
fluctuations (on a set of fixed trajectory segments) from the average heat
entropy absorbed in thermostat were found to be non-Gaussian. Approximately,
the fluctuations can be discribed by a two-Gaussian distribution with a
crossover independent of the segment length and the number of trajectories
('particles'). The distribution itself does depend on both, approaching the
single standard Gaussian distribution as any of those parameters increases. The
global time-dependent fluctuations turned out to be qualitatively different in
that they have a strict upper bound much less than the average entropy
production. Thus, unlike the equilibrium steady state, the recovery of the
initial low entropy becomes impossible, after a sufficiently long time, even in
the largest fluctuations. However, preliminary numerical experiments and the
theoretical estimates in the special case of the critical dynamics with
superdiffusion suggest the existence of infinitely many Poincar\'e recurrences
to the initial state and beyond. This is a new interesting phenomenon to be
farther studied together with some other open questions. Relation of this
particular example of nonequilibrium steady state to a long-standing persistent
controversy over statistical 'irreversibility', or the notorious 'time arrow',
is also discussed. In conclusion, an unsolved problem of the origin of the
causality 'principle' is touched upon. | nlin_CD |
Detecting Generalized Synchronization Between Chaotic Signals: A
Kernel-based Approach: A unified framework for analyzing generalized synchronization in coupled
chaotic systems from data is proposed. The key of the proposed approach is the
use of the kernel methods recently developed in the field of machine learning.
Several successful applications are presented, which show the capability of the
kernel-based approach for detecting generalized synchronization. It is also
shown that the dynamical change of the coupling coefficient between two chaotic
systems can be captured by the proposed approach. | nlin_CD |
Entropy production and Lyapunov instability at the onset of turbulent
convection: Computer simulations of a compressible fluid, convecting heat in two
dimensions, suggest that, within a range of Rayleigh numbers, two distinctly
different, but stable, time-dependent flow morphologies are possible. The
simpler of the flows has two characteristic frequencies: the rotation frequency
of the convecting rolls, and the vertical oscillation frequency of the rolls.
Observables, such as the heat flux, have a simple-periodic (harmonic) time
dependence. The more complex flow has at least one additional characteristic
frequency -- the horizontal frequency of the cold, downward- and the warm,
upward-flowing plumes. Observables of this latter flow have a broadband
frequency distribution. The two flow morphologies, at the same Rayleigh number,
have different rates of entropy production and different Lyapunov exponents.
The simpler "harmonic" flow transports more heat (produces entropy at a greater
rate), whereas the more complex "chaotic" flow has a larger maximum Lyapunov
exponent (corresponding to a larger rate of phase-space information loss). A
linear combination of these two rates is invariant for the two flow
morphologies over the entire range of Rayleigh numbers for which the flows
coexist, suggesting a relation between the two rates near the onset of
convective turbulence. | nlin_CD |
Timing of Transients: Quantifying Reaching Times and Transient Behavior
in Complex Systems: When quantifying the time spent in the transient of a complex dynamical
system, the fundamental problem is that for a large class of systems the actual
time for reaching an attractor is infinite. Common methods for dealing with
this problem usually introduce three additional problems: non-invariance,
physical interpretation, and discontinuities, calling for carefully designed
methods for quantifying transients.
In this article, we discuss how the aforementioned problems emerge and
propose two novel metrics, Regularized Reaching Time ($T_{RR}$) and Area under
Distance Curve (AUDIC), to solve them, capturing two complementary aspects of
the transient dynamics.
$T_{RR}$ quantifies the additional time (positive or negative) that a
trajectory starting at a chosen initial condition needs to reach the attractor
after a reference trajectory has already arrived there. A positive or negative
value means that it arrives by this much earlier or later than the reference.
Because $T_{RR}$ is an analysis of return times after shocks, it is a
systematic approach to the concept of critical slowing down [1]; hence it is
naturally an early-warning signal [2] for bifurcations when central statistics
over distributions of initial conditions are used.
AUDIC is the distance of the trajectory to the attractor integrated over
time. Complementary to $T_{RR}$, it measures which trajectories are reluctant,
i.e. stay away from the attractor for long, or eager to approach it right away.
(... shortened for arxiv listing, full abstract in paper ...) New features in
these models can be uncovered, including the surprising regularity of the
Roessler system's basin of attraction even in the regime of a chaotic
attractor. Additionally, we demonstrate the critical slowing down
interpretation by presenting the metrics' sensitivity to prebifurcational
change and thus how they act as early-warning signals. | nlin_CD |
Intensity distribution of non-linear scattering states: We investigate the interplay between coherent effects characteristic of the
propagation of linear waves, the non-linear effects due to interactions, and
the quantum manifestations of classical chaos due to geometrical confinement,
as they arise in the context of the transport of Bose-Einstein condensates. We
specifically show that, extending standard methods for non-interacting systems,
the body of the statistical distribution of intensities for scattering states
solving the Gross-Pitaevskii equation is very well described by a local
Gaussian ansatz with a position-dependent variance. We propose a semiclassical
approach based on interfering classical paths to fix the single parameter
describing the universal deviations from a global Gaussian distribution. Being
tail effects, rare events like rogue waves characteristic of non-linear field
equations do not affect our results. | nlin_CD |
A Plethora of Strange Nonchaotic Attractors: We show that it is possible to devise a large class of skew--product
dynamical systems which have strange nonchaotic attractors (SNAs): the dynamics
is asymptotically on fractal attractors and the largest Lyapunov exponent is
nonpositive. Furthermore, we show that quasiperiodic forcing, which has been a
hallmark of essentially allhitherto known examples of such dynamics is {\it
not} necessary for the creation of SNAs. | nlin_CD |
Stochastic mean field formulation of the dynamics of diluted neural
networks: We consider pulse-coupled Leaky Integrate-and-Fire neural networks with
randomly distributed synaptic couplings. This random dilution induces
fluctuations in the evolution of the macroscopic variables and deterministic
chaos at the microscopic level. Our main aim is to mimic the effect of the
dilution as a noise source acting on the dynamics of a globally coupled
non-chaotic system. Indeed, the evolution of a diluted neural network can be
well approximated as a fully pulse coupled network, where each neuron is driven
by a mean synaptic current plus additive noise. These terms represent the
average and the fluctuations of the synaptic currents acting on the single
neurons in the diluted system. The main microscopic and macroscopic dynamical
features can be retrieved with this stochastic approximation. Furthermore, the
microscopic stability of the diluted network can be also reproduced, as
demonstrated from the almost coincidence of the measured Lyapunov exponents in
the deterministic and stochastic cases for an ample range of system sizes. Our
results strongly suggest that the fluctuations in the synaptic currents are
responsible for the emergence of chaos in this class of pulse coupled networks. | nlin_CD |
Unraveling the Chaos-land and its organization in the Rabinovich System: A suite of analytical and computational techniques based on symbolic
representations of simple and complex dynamics, is further developed and
employed to unravel the global organization of bi-parametric structures that
underlie the emergence of chaos in a simplified resonantly coupled wave triplet
system, known as the Rabinovich system. Bi-parametric scans reveal the stunning
intricacy and intramural connections between homoclinic and heteroclinic
connections, and codimension-2 Bykov T-points and saddle structures, which are
the prime organizing centers of complexity of the bifurcation unfolding of the
given system. This suite includes Deterministic Chaos Prospector (DCP) to sweep
and effectively identify regions of simple (Morse-Smale) and chaotic
structurally unstable dynamics in the system. Our analysis provides striking
new insights into the complex behaviors exhibited by this and similar systems. | nlin_CD |
A Non-Equilibrium Defect-Unbinding Transition: Defect Trajectories and
Loop Statistics: In a Ginzburg-Landau model for parametrically driven waves a transition
between a state of ordered and one of disordered spatio-temporal defect chaos
is found. To characterize the two different chaotic states and to get insight
into the break-down of the order, the trajectories of the defects are tracked
in detail. Since the defects are always created and annihilated in pairs the
trajectories form loops in space time. The probability distribution functions
for the size of the loops and the number of defects involved in them undergo a
transition from exponential decay in the ordered regime to a power-law decay in
the disordered regime. These power laws are also found in a simple lattice
model of randomly created defect pairs that diffuse and annihilate upon
collision. | nlin_CD |
Designing two-dimensional limit-cycle oscillators with prescribed
trajectories and phase-response characteristics: We propose a method for designing two-dimensional limit-cycle oscillators
with prescribed periodic trajectories and phase response properties based on
the phase reduction theory, which gives a concise description of
weakly-perturbed limit-cycle oscillators and is widely used in the analysis of
synchronization dynamics. We develop an algorithm for designing the vector
field with a stable limit cycle, which possesses a given shape and also a given
phase sensitivity function. The vector field of the limit-cycle oscillator is
approximated by polynomials whose coefficients are estimated by convex
optimization. Linear stability of the limit cycle is ensured by introducing an
upper bound to the Floquet exponent. The validity of the proposed method is
verified numerically by designing several types of two-dimensional existing and
artificial oscillators. As applications, we first design a limit-cycle
oscillator with an artificial star-shaped periodic trajectory and demonstrate
global entrainment. We then design a limit-cycle oscillator with an artificial
high-harmonic phase sensitivity function and demonstrate multistable
entrainment caused by a high-frequency periodic input. | nlin_CD |
Statistical and dynamical properties of the quantum triangle map: We study the statistical and dynamical properties of the quantum triangle
map, whose classical counterpart can exhibit ergodic and mixing dynamics, but
is never chaotic. Numerical results show that ergodicity is a sufficient
condition for spectrum and eigenfunctions to follow the prediction of Random
Matrix Theory, even though the underlying classical dynamics is not chaotic. On
the other hand, dynamical quantities such as the out-of-time-ordered correlator
(OTOC) and the number of harmonics, exhibit a growth rate vanishing in the
semiclassical limit, in agreement with the fact that classical dynamics has
zero Lyapunov exponent. Our finding show that, while spectral statistics can be
used to detect ergodicity, OTOC and number of harmonics are diagnostics of
chaos. | nlin_CD |
Scaling regimes of 2d turbulence with power law stirring: theories
versus numerical experiments: We inquire the statistical properties of the pair formed by the Navier-Stokes
equation for an incompressible velocity field and the advection-diffusion
equation for a scalar field transported in the same flow in two dimensions
(2d). The system is in a regime of fully developed turbulence stirred by
forcing fields with Gaussian statistics, white-noise in time and self-similar
in space. In this setting and if the stirring is concentrated at small spatial
scales as if due to thermal fluctuations, it is possible to carry out a
first-principle ultra-violet renormalization group analysis of the scaling
behavior of the model.
Kraichnan's phenomenological theory of two dimensional turbulence upholds the
existence of an inertial range characterized by inverse energy transfer at
scales larger than the stirring one. For our model Kraichnan's theory, however,
implies scaling predictions radically discordant from the renormalization group
results. We perform accurate numerical experiments to assess the actual
statistical properties of 2d-turbulence with power-law stirring. Our results
clearly indicate that an adapted version of Kraichnan's theory is consistent
with the observed phenomenology. We also provide some theoretical scenarios to
account for the discrepancy between renormalization group analysis and the
observed phenomenology. | nlin_CD |
Long-time signatures of short-time dynamics in decaying quantum-chaotic
systems: We analyze the decay of classically chaotic quantum systems in the presence
of fast ballistic escape routes on the Ehrenfest time scale. For a continuous
excitation process, the form factor of the decay cross section deviates from
the universal random-matrix result on the Heisenberg time scale, i.e. for times
much larger than the time for ballistic escape. We derive an exact analytical
description and compare our results with numerical simulations for a dynamical
model. | nlin_CD |
A Phase-Space Approach for Propagating Field-Field Correlation Functions: We show that radiation from complex and inherently random but correlated wave
sources can be modelled efficiently by using an approach based on the Wigner
distribution function. Our method exploits the connection between correlation
functions and theWigner function and admits in its simplest approximation a
direct representation in terms of the evolution of ray densities in phase
space. We show that next leading order corrections to the ray-tracing
approximation lead to Airy-function type phase space propagators. By exploiting
the exact Wigner function propagator, inherently wave-like effects such as
evanescent decay or radiation from more heterogeneous sources as well as
diffraction and reflections can be included and analysed. We discuss in
particular the role of evanescent waves in the near-field of non-paraxial
sources and give explicit expressions for the growth rate of the correlation
length as function of the distance from the source. Furthermore, results for
the reflection of partially coherent sources from flat mirrors are given. We
focus here on electromagnetic sources at microwave frequencies and modelling
efforts in the context of electromagnetic compatibility. | nlin_CD |
The dynamical temperature and the standard map: Numerical experiments with the standard map at high values of the
stochasticity parameter reveal the existence of simple analytical relations
connecting the volume and the dynamical temperature of the chaotic component of
the phase space. | nlin_CD |
Synchronized bursts following instability of synchronous spiking in
chaotic neuronal networks: We report on the origin of synchronized bursting dynamics in various networks
of neural spiking oscillators, when a certain threshold in coupling strength is
exceeded. These ensembles synchronize at relatively low coupling strength and
lose synchronization at stronger coupling via spatio-temporal intermittency.
The latter transition triggers multiple-timescale dynamics, which results in
synchronized bursting with a fractal-like spatio-temporal pattern of spiking.
Implementation of an appropriate technique of separating oscillations on
different time-scales allows for quantitative analysis of this phenomenon. We
show, that this phenomenon is generic for various network topologies from
regular to small-world and scale-free ones and for different types of coupling. | nlin_CD |
Excitable systems with noise and delay with applications to control:
renewal theory approach: We present an approach for the analytical treatment of excitable systems with
noise-induced dynamics in the presence of time delay. An excitable system is
modeled as a bistable system with a time delay, while another delay enters as a
control term taken after [Pyragas 1992] as a difference between the current
system state and its state "tau" time units before. This approach combines the
elements of renewal theory to estimate the essential features of the resulting
stochastic process as functions of the parameters of the controlling term. | nlin_CD |
Noise-enhanced trapping in chaotic scattering: We show that noise enhances the trapping of trajectories in scattering
systems. In fully chaotic systems, the decay rate can decrease with increasing
noise due to a generic mismatch between the noiseless escape rate and the value
predicted by the Liouville measure of the exit set. In Hamiltonian systems with
mixed phase space we show that noise leads to a slower algebraic decay due to
trajectories performing a random walk inside Kolmogorov-Arnold-Moser islands.
We argue that these noise-enhanced trapping mechanisms exist in most scattering
systems and are likely to be dominant for small noise intensities, which is
confirmed through a detailed investigation in the Henon map. Our results can be
tested in fluid experiments, affect the fractal Weyl's law of quantum systems,
and modify the estimations of chemical reaction rates based on phase-space
transition state theory. | nlin_CD |
The paradox of infinitesimal granularity: Chaos and the reversibility of
time in Newton's theory of gravity: The fundamental laws of physics are time-symmetric, but our macroscopic
experience contradicts this. The time reversibility paradox is partly a
consequence of the unpredictability of Newton's equations of motion. We measure
the dependence of the fraction of irreversible, gravitational N-body systems on
numerical precision and find that it scales as a power law. The stochastic wave
packet reduction postulate then introduces fundamental uncertainties in the
Cartesian phase space coordinates that propagate through classical three-body
dynamics to macroscopic scales within the triple's lifetime. The spontaneous
collapse of the wave function then drives the global chaotic behavior of the
Universe through the superposition of triple systems (and probably multi-body
systems). The paradox of infinitesimal granularity then arises from the
superposition principle, which states that any multi-body system is composed of
an ensemble of three-body problems. | nlin_CD |
Functional renormalization-group approach to decaying turbulence: We reconsider the functional renormalization-group (FRG) approach to decaying
Burgers turbulence, and extend it to decaying Navier-Stokes and
Surface-Quasi-Geostrophic turbulence. The method is based on a renormalized
small-time expansion, equivalent to a loop expansion, and naturally produces a
dissipative anomaly and a cascade after a finite time. We explicitly calculate
and analyze the one-loop FRG equations in the zero-viscosity limit as a
function of the dimension. For Burgers they reproduce the FRG equation obtained
in the context of random manifolds, extending previous results of one of us.
Breakdown of energy conservation due to shocks and the appearance of a direct
energy cascade corresponds to failure of dimensional reduction in the context
of disordered systems. For Navier-Stokes in three dimensions, the
velocity-velocity correlation function acquires a linear dependence on the
distance, zeta_2=1, in the inertial range, instead of Kolmogorov's zeta_2=2/3;
however the possibility remains for corrections at two- or higher-loop order.
In two dimensions, we obtain a numerical solution which conserves energy and
exhibits an inverse cascade, with explicit analytical results both for large
and small distances, in agreement with the scaling proposed by Batchelor. In
large dimensions, the one-loop FRG equation for Navier-Stokes converges to that
of Burgers. | nlin_CD |
Stabilisation of long-period periodic orbits using time-delayed feedback
control: The Pyragas method of feedback control has attracted much interest as a
method of stabilising unstable periodic orbits in a number of situations. We
show that a time-delayed feedback control similar to the Pyragas method can be
used to stabilise periodic orbits with arbitrarily large period, specifically
those resulting from a resonant bifurcation of a heteroclinic cycle. Our
analysis reduces the infinite-dimensional delay-equation governing the system
with feedback to a three-dimensional map, by making certain assumptions about
the form of the solutions. The stability of a fixed point in this map
corresponds to the stability of the periodic orbit in the flow, and can be
computed analytically. We compare the analytic results to a numerical example
and find very good agreement. | nlin_CD |
A Mechanical Analog of the Two-bounce Resonance of Solitary Waves:
Modeling and Experiment: We describe a simple mechanical system, a ball rolling along a
specially-designed landscape, that mimics the dynamics of a well known
phenomenon, the two-bounce resonance of solitary wave collisions, that has been
seen in countless numerical simulations but never in the laboratory. We provide
a brief history of the solitary wave problem, stressing the fundamental role
collective-coordinate models played in understanding this phenomenon. We derive
the equations governing the motion of a point particle confined to such a
surface and then design a surface on which to roll the ball, such that its
motion will evolve under the same equations that approximately govern solitary
wave collisions. We report on physical experiments, carried out in an
undergraduate applied mathematics course, that seem to verify one aspect of
chaotic scattering, the so-called two-bounce resonance. | nlin_CD |
Simple models of bouncing ball dynamics and their comparison: Nonlinear dynamics of a bouncing ball moving in gravitational field and
colliding with a moving limiter is considered. Several simple models of table
motion are studied and compared. Dependence of displacement of the table on
time, approximating sinusoidal motion and making analytical computations
possible, is assumed as quadratic and cubic functions of time, respectively. | nlin_CD |
Intermittency effects in Burgers equation driven by thermal noise: For the Burgers equation driven by thermal noise leading asymptotics of pair
and high-order correlators of the velocity field are found for finite times and
large distances. It is shown that the intermittency takes place: some
correlators are much larger than their reducible parts. | nlin_CD |
Morphological Image Analysis of Quantum Motion in Billiards: Morphological image analysis is applied to the time evolution of the
probability distribution of a quantum particle moving in two and
three-dimensional billiards. It is shown that the time-averaged Euler
characteristic of the probability density provides a well defined quantity to
distinguish between classically integrable and non-integrable billiards. In
three dimensions the time-averaged mean breadth of the probability density may
also be used for this purpose. | nlin_CD |
From quasiperiodicity to high-dimensional chaos without intermediate
low-dimensional chaos: We study and characterize a direct route to high-dimensional chaos (i.e. not
implying an intermediate low-dimensional attractor) of a system composed out of
three coupled Lorenz oscillators. A geometric analysis of this
medium-dimensional dynamical system is carried out through a variety of
numerical quantitative and qualitative techniques, that ultimately lead to the
reconstruction of the route. The main finding is that the transition is
organized by a heteroclinic explosion. The observed scenario resembles the
classical route to chaos via homoclinic explosion of the Lorenz model. | nlin_CD |
On the Implementation of the 0-1 Test for Chaos: In this paper we address practical aspects of the implementation of the 0-1
test for chaos in deterministic systems. In addition, we present a new
formulation of the test which significantly increases its sensitivity. The test
can be viewed as a method to distill a binary quantity from the power spectrum.
The implementation is guided by recent results from the theoretical
justification of the test as well as by exploring better statistical methods to
determine the binary quantities. We give several examples to illustrate the
improvement. | nlin_CD |
General mechanism for amplitude death in coupled systems: We introduce a general mechanism for amplitude death in coupled
synchronizable dynamical systems. It is known that when two systems are coupled
directly, they can synchronize under suitable conditions. When an indirect
feedback coupling through an environment or an external system is introduced in
them, it is found to induce a tendency for anti-synchronization. We show that,
for sufficient strengths, these two competing effects can lead to amplitude
death. We provide a general stability analysis that gives the threshold values
for onset of amplitude death. We study in detail the nature of the transition
to death in several specific cases and find that the transitions can be of two
types - continuous and discontinuous. By choosing a variety of dynamics for
example, periodic, chaotic, hyper chaotic, and time-delay systems, we
illustrate that this mechanism is quite general and works for different types
of direct coupling, such as diffusive, replacement, and synaptic couplings and
for different damped dynamics of the environment. | nlin_CD |
Fluctuation-response relation in turbulent systems: We address the problem of measuring time-properties of Response Functions
(Green functions) in Gaussian models (Orszag-McLaughin) and strongly
non-Gaussian models (shell models for turbulence). We introduce the concept of
{\it halving time statistics} to have a statistically stable tool to quantify
the time decay of Response Functions and Generalized Response Functions of high
order. We show numerically that in shell models for three dimensional
turbulence Response Functions are inertial range quantities. This is a strong
indication that the invariant measure describing the shell-velocity
fluctuations is characterized by short range interactions between neighboring
shells. | nlin_CD |
Non-smooth model and numerical analysis of a friction driven structure
for piezoelectric motors: In this contribution, typical friction driven structures are summarized and
presented considering the mechanical structures and operation principles of
different types of piezoelectric motors. A two degree-of-freedom dynamic model
with one unilateral frictional contact is built for one of the friction driven
structures. Different contact regimes and the transitions between them are
identified and analyzed. Numerical simulations are conducted to find out
different operation modes of the system concerning the sequence of contact
regimes in one steady state period. The influences of parameters on the
operation modes and corresponding steady state characteristics are also
explored. Some advice are then given in terms of the design of friction driven
structures and piezoelectric motors. | nlin_CD |
Chaos suppression in the parametrically driven Lorenz system: We predict theoretically and verify experimentally the suppression of chaos
in the Lorenz system driven by a high-frequency periodic or stochastic
parametric force. We derive the theoretical criteria for chaos suppression and
verify that they are in a good agreement with the results of numerical
simulations and the experimental data obtained for an analog electronic
circuit. | nlin_CD |
Coexisting synchronous and asynchronous states in locally coupled array
of oscillators by partial self-feedback control: We report the emergence of coexisting synchronous and asynchronous
subpopulations of oscillators in one dimensional arrays of identical
oscillators by applying a self-feedback control. When a self-feedback is
applied to a subpopulation of the array, similar to chimera states, it splits
into two/more sub-subpopulations coexisting in coherent and incoherent states
for a range of self-feedback strength. By tuning the coupling between the
nearest neighbors and the amount of self-feedback in the perturbed
subpopulation, the size of the coherent and the incoherent sub-subpopulations
in the array can be controlled, although the exact size of them is
unpredictable. We present numerical evidence using the Landau-Stuart (LS)
system and the Kuramoto-Sakaguchi (KS) phase model. | nlin_CD |
Dynamic Phase Transition from Localized to Spatiotemporal Chaos in
Coupled Circle Map with Feedback: We investigate coupled circle maps in presence of feedback and explore
various dynamical phases observed in this system of coupled high dimensional
maps. We observe an interesting transition from localized chaos to
spatiotemporal chaos. We study this transition as a dynamic phase transition.
We observe that persistence acts as an excellent quantifier to describe this
transition. Taking the location of the fixed point of circle map (which does
not change with feedback) as a reference point, we compute number of sites
which have been greater than (less than) the fixed point till time t. Though
local dynamics is high-dimensional in this case this definition of persistence
which tracks a single variable is an excellent quantifier for this transition.
In most cases, we also obtain a well defined persistence exponent at the
critical point and observe conventional scaling as seen in second order phase
transitions. This indicates that persistence could work as good order parameter
for transitions from fully or partially arrested phase. We also give an
explanation of gaps in eigenvalue spectrum of the Jacobian of localized state. | nlin_CD |
On WKB Series for the Radial Kepler Problem: We obtain the rigorous WKB expansion to all orders for the radial Kepler
problem, using the residue calculus in evaluating the WKB quantization
condition in terms of a complex contour integral in the complexified coordinate
plane. The procedure yields the exact energy spectrum of this Schr\"odinger
eigenvalue problem and thus resolves the controversies around the so-called
"Langer correction". The problem is nontrivial also because there are only a
few systems for which all orders of the WKB series can be calculated, yielding
a convergent series whose sum is equal to the exact result, and thus sheds new
light to similar and more difficult problems. | nlin_CD |
The reflection-antisymmetric counterpart of the Kármán-Howarth
dynamical equation: We study the isotropic, helical component in homogeneous turbulence using
statistical objects which have the correct symmetry and parity properties.
Using these objects we derive an analogue of the K\'arm\'an-Howarth equation,
that arises due to parity violation in isotropic flows. The main equation we
obtain is consistent with the results of O. Chkhetiani [JETP, 63, 768, (1996)]
and
V.S. L'vov et al. [chao-dyn/9705016,
(1997)] but is derived using only velocity correlations, with no direct
consideration of the vorticity or helicity. This alternative formulation offers
an advantage to both experimental and numerical measurements. We also
postulate, under the assumption of self-similarity, the existence of a
hierarchy of scaling exponents for helical velocity correlation functions of
arbitrary order, analogous to the
Kolmogorov 1941 prediction for the scaling exponents of velocity structure
function. | nlin_CD |
Chirikov and Nekhoroshev diffusion estimates: bridging the two sides of
the river: We present theoretical and numerical results pointing towards a strong
connection between the estimates for the diffusion rate along simple resonances
in multidimensional nonlinear Hamiltonian systems that can be obtained using
the heuristic theory of Chirikov and a more formal one due to Nekhoroshev. We
show that, despite a wide-spread impression, the two theories are complementary
rather than antagonist. Indeed, although Chirikov's 1979 review has thousands
of citations, almost all of them refer to topics such as the resonance overlap
criterion, fast diffusion, the Standard or Whisker Map, and not to the
constructive theory providing a formula to measure diffusion along a single
resonance. However, as will be demonstrated explicitly below, Chirikov's
formula provides values of the diffusion coefficient which are quite well
comparable to the numerically computed ones, provided that it is implemented on
the so-called optimal normal form derived as in the analytic part of
Nekhoroshev's theorem. On the other hand, Chirikov's formula yields unrealistic
values of the diffusion coefficient, in particular for very small values of the
perturbation, when used in the original Hamiltonian instead of the optimal
normal form. In the present paper, we take advantage of this complementarity in
order to obtain accurate theoretical predictions for the local value of the
diffusion coefficient along a resonance in a specific 3DoF nearly integrable
Hamiltonian system. Besides, we compute numerically the diffusion coefficient
and a full comparison of all estimates is made for ten values of the
perturbation parameter, showing a very satisfactory agreement. | nlin_CD |
A study of the double pendulum using polynomial optimization: In dynamical systems governed by differential equations, a guarantee that
trajectories emanating from a given set of initial conditions do not enter
another given set can be obtained by constructing a barrier function that
satisfies certain inequalities on phase space. Often these inequalities amount
to nonnegativity of polynomials and can be enforced using sum-of-squares
conditions, in which case barrier functions can be constructed computationally
using convex optimization over polynomials. To study how well such computations
can characterize sets of initial conditions in a chaotic system, we use the
undamped double pendulum as an example and ask which stationary initial
positions do not lead to flipping of the pendulum within a chosen time window.
Computations give semialgebraic sets that are close inner approximations to the
fractal set of all such initial positions. | nlin_CD |
Defining Chaos: In this paper we propose, discuss and illustrate a computationally feasible
definition of chaos which can be applied very generally to situations that are
commonly encountered, including attractors, repellers and non-periodically
forced systems. This definition is based on an entropy-like quantity, which we
call "expansion entropy", and we define chaos as occurring when this quantity
is positive. We relate and compare expansion entropy to the well-known concept
of topological entropy, to which it is equivalent under appropriate conditions.
We also present example illustrations, discuss computational implementations,
and point out issues arising from attempts at giving definitions of chaos that
are not entropy-based. | nlin_CD |
Alpha-modeling strategy for LES of turbulent mixing: The $\alpha$-modeling strategy is followed to derive a new subgrid
parameterization of the turbulent stress tensor in large-eddy simulation (LES).
The LES-$\alpha$ modeling yields an explicitly filtered subgrid
parameterization which contains the filtered nonlinear gradient model as well
as a model which represents `Leray-regularization'. The LES-$\alpha$ model is
compared with similarity and eddy-viscosity models that also use the dynamic
procedure. Numerical simulations of a turbulent mixing layer are performed
using both a second order, and a fourth order accurate finite volume
discretization. The Leray model emerges as the most accurate, robust and
computationally efficient among the three LES-$\alpha$ subgrid
parameterizations for the turbulent mixing layer. The evolution of the resolved
kinetic energy is analyzed and the various subgrid-model contributions to it
are identified. By comparing LES-$\alpha$ at different subgrid resolutions, an
impression of finite volume discretization error dynamics is obtained. | nlin_CD |
Dynamical complexity as a proxy for the network degree distribution: We explore the relation between the topological relevance of a node in a
complex network and the individual dynamics it exhibits. When the system is
weakly coupled, the effect of the coupling strength against the dynamical
complexity of the nodes is found to be a function of their topological role,
with nodes of higher degree displaying lower levels of complexity. We provide
several examples of theoretical models of chaotic oscillators, pulse-coupled
neurons and experimental networks of nonlinear electronic circuits evidencing
such a hierarchical behavior. Importantly, our results imply that it is
possible to infer the degree distribution of a network only from individual
dynamical measurements. | nlin_CD |
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