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Basins of attraction of equilibrium points in the planar circular
restricted five-body problem: We numerically explore the Newton-Raphson basins of convergence, related to
the libration points (which act as attractors), in the planar circular
restricted five-body problem (CR5BP). The evolution of the position and the
linear stability of the equilibrium points is determined, as a function of the
value of the mass parameter. The attracting regions, on several types of two
dimensional planes, are revealed by using the multivariate version of the
classical Newton-Raphson iterative method. We perform a systematic
investigation in an attempt to understand how the mass parameter affects the
geometry as well as the degree of fractality of the basins of attraction. The
regions of convergence are also related with the required number of iterations
and also with the corresponding probability distributions. | nlin_CD |
Transport properties in chaotic and non-chaotic many particles systems: Two deterministic models for Brownian motion are investigated by means of
numerical simulations and kinetic theory arguments. The first model consists of
a heavy hard disk immersed in a rarefied gas of smaller and lighter hard disks
acting as a thermal bath. The second is the same except for the shape of the
particles, which is now square. The basic difference of these two systems lies
in the interaction: hard core elastic collisions make the dynamics of the disks
chaotic whereas that of squares is not. Remarkably, this difference is not
reflected in the transport properties of the two systems: simulations show that
the diffusion coefficients, velocity correlations and response functions of the
heavy impurity are in agreement with kinetic theory for both the chaotic and
the non-chaotic model. The relaxation to equilibrium, however, is very
sensitive to the kind of interaction. These observations are used to reconsider
and discuss some issues connected to chaos, statistical mechanics and
diffusion. | nlin_CD |
Intermittent stickiness synchronization: This work uses the statistical properties of Finite-Time Lyapunov Exponents
(FTLEs) to investigate the Intermittent Stickiness Synchronization (ISS)
observed in the mixed phase space of high-dimensional Hamiltonian systems. Full
Stickiness Synchronization (SS) occurs when all FTLEs from a chaotic trajectory
tend to zero for arbitrary long time-windows. This behavior is a consequence of
the sticky motion close to regular structures which live in the
high-dimensional phase space and affects all unstable directions proportionally
by the same amount, generating a kind of collective motion. Partial SS occurs
when at least one FTLE approaches to zero. Thus, distinct degrees of partial SS
may occur, depending on the values of nonlinearity and coupling parameters, on
the dimension of the phase space and on the number of positive FTLEs. Through
filtering procedures used to precisely characterize the sticky motion, we are
able to compute the algebraic decay exponents of the ISS and to obtain
remarkable evidence about the existence of a universal behavior related to the
decay of time correlations encoded in such exponents. In addition, we show that
even though the probability to find full SS is small compared to partial SSs,
the full SS may appear for very long times due to the slow algebraic decay of
time correlations in mixed phase space. In this sense, observations of very
late intermittence between chaotic motion and full SSs become rare events. | nlin_CD |
On Turbulence of Polymer Solutions: We investigate high-Reynolds number turbulence in dilute polymer solutions.
We show the existence of a critical value of the Reynolds number which
separates two different regimes. In the first regime, below the transition, the
influence of the polymer molecules on the flow is negligible and they can be
regarded as passively embedded in the flow. This case admits a detailed
investigation of the statistics of the polymer elongations. The second state is
realized when the Reynolds number is larger than the critical value. This
regime is characterized by the strong back reaction of polymers on the flow. We
establish some properties of the statistics of the stress and velocity in this
regime and discuss its relation to the drag reduction phenomenon. | nlin_CD |
The Fermi-Pasta-Ulam problem: 50 years of progress: A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with
its suggested resolutions and its relation to other physical problems. We focus
on the ideas and concepts that have become the core of modern nonlinear
mechanics, in their historical perspective. Starting from the first numerical
results of FPU, both theoretical and numerical findings are discussed in close
connection with the problems of ergodicity, integrability, chaos and stability
of motion. New directions related to the Bose-Einstein condensation and quantum
systems of interacting Bose-particles are also considered. | nlin_CD |
Random global coupling induces synchronization and nontrivial collective
behavior in networks of chaotic maps: The phenomena of synchronization and nontrivial collective behavior are
studied in a model of coupled chaotic maps with random global coupling. The
mean field of the system is coupled to a fraction of elements randomly chosen
at any given time. It is shown that the reinjection of the mean field to a
fraction of randomly selected elements can induce synchronization and
nontrivial collective behavior in the system. The regions where these
collective states emerge on the space of parameters of the system are
calculated. | nlin_CD |
Restoration of oscillation in network of oscillators in presence of
direct and indirect interactions: The suppression of oscillations in coupled systems may lead to several
unwanted situations, which requires a suitable treatment to overcome the
suppression. In this paper, we show that the environmental coupling in the
presence of direct interaction, which can suppress oscillation even in a
network of identical oscillators, can be modified by introducing a feedback
factor in the coupling scheme in order to restore the oscillation. We inspect
how the introduction of the feedback factor helps to resurrect oscillation from
various kind of death states. We numerically verify the resurrection of
oscillations for two paradigmatic limit cycle systems, namely Landau-Stuart and
Van der Pol oscillators and also in generic chaotic Lorenz oscillator. We also
study the effect of parameter mismatch in the process of restoring oscillation
for coupled oscillators. | nlin_CD |
Nonlocal mechanism for cluster synchronization in neural circuits: The interplay between the topology of cortical circuits and synchronized
activity modes in distinct cortical areas is a key enigma in neuroscience. We
present a new nonlocal mechanism governing the periodic activity mode: the
greatest common divisor (GCD) of network loops. For a stimulus to one node, the
network splits into GCD-clusters in which cluster neurons are in zero-lag
synchronization. For complex external stimuli, the number of clusters can be
any common divisor. The synchronized mode and the transients to synchronization
pinpoint the type of external stimuli. The findings, supported by an
information mixing argument and simulations of Hodgkin Huxley population
dynamic networks with unidirectional connectivity and synaptic noise, call for
reexamining sources of correlated activity in cortex and shorter information
processing time scales. | nlin_CD |
Control and Synchronization of Chaotic Fractional-Order Coullet System
via Active Controller: In this paper, fractional order Coullet system is studied. An active control
technique is applied to control this chaotic system. This type of controller is
also applied to synchronize chaotic fractional-order systems in master-slave
structure. The synchronization procedure is shown via simulation. The boundary
of stability is obtained by both of theoretical analysis and simulation result.
The numerical simulations show the effectiveness of the proposed controller. | nlin_CD |
Synchronization in hyperchaotic time-delayed electronic oscillators
coupled indirectly via a common environment: The present paper explores the synchronization scenario of hyperchaotic
time-delayed electronic oscillators coupled indirectly via a common
environment. We show that depending upon the coupling parameters a hyperchaotic
time-delayed system can show in-phase or complete synchronization, and also
inverse-phase or anti-synchronization. This paper reports the first
experimental confirmation of synchronization of hyperchaos in time-delayed
electronic oscillators coupled indirectly through a common environment. We
confirm the occurrence of in-phase and inverse-phase synchronization phenomena
in the coupled system through the dynamical measures like generalized
autocorrelation function, correlation of probability of recurrence, and the
concept of localized sets computed directly from the experimental time-series
data. We also present a linear stability analysis of the coupled system. The
experimental and analytical results are further supported by the detailed
numerical analysis of the coupled system. Apart from the above mentioned
measures, we numerically compute another quantitative measure, namely, Lyapunov
exponent spectrum of the coupled system that confirms the transition from the
in-phase (inverse-phase) synchronized state to the complete (anti-)
synchronized state with the increasing coupling strength. | nlin_CD |
On the Thermodynamic Formalism for the Farey Map: The chaotic phenomenon of intermittency is modeled by a simple map of the
unit interval, the Farey map. The long term dynamical behaviour of a point
under iteration of the map is translated into a spin system via symbolic
dynamics. Methods from dynamical systems theory and statistical mechanics may
then be used to analyse the map, respectively the zeta function and the
transfer operator. Intermittency is seen to be problematic to analyze due to
the presence of an `indifferent fixed point'. Points under iteration of the map
move away from this point extremely slowly creating pathological convergence
times for calculations. This difficulty is removed by going to an appropriate
induced subsystem, which also leads to an induced zeta function and an induced
transfer operator. Results obtained there can be transferred back to the
original system. The main work is then divided into two sections. The first
demonstrates a connection between the induced versions of the zeta function and
the transfer operator providing useful results regarding the analyticity of the
zeta function. The second section contains a detailed analysis of the pressure
function for the induced system and hence the original by considering bounds on
the radius of convergence of the induced zeta function. In particular, the
asymptotic behaviour of the pressure function in the limit $\beta$, the inverse
of `temperature', tends to negative infinity is determined and the existence
and nature of a phase transition at $\beta=1$ is also discussed. | nlin_CD |
Semiclassical Trace Formulas for Noninteracting Identical Particles: We extend the Gutzwiller trace formula to systems of noninteracting identical
particles. The standard relation for isolated orbits does not apply since the
energy of each particle is separately conserved causing the periodic orbits to
occur in continuous families. The identical nature of the particles also
introduces discrete permutational symmetries. We exploit the formalism of
Creagh and Littlejohn [Phys. Rev. A 44, 836 (1991)], who have studied
semiclassical dynamics in the presence of continuous symmetries, to derive
many-body trace formulas for the full and symmetry-reduced densities of states.
Numerical studies of the three-particle cardioid billiard are used to
explicitly illustrate and test the results of the theory. | nlin_CD |
Analysing Lyapunov spectra of chaotic dynamical systems: It is shown that the asymptotic spectra of finite-time Lyapunov exponents of
a variety of fully chaotic dynamical systems can be understood in terms of a
statistical analysis. Using random matrix theory we derive numerical and in
particular analytical results which provide insights into the overall behaviour
of the Lyapunov exponents particularly for strange attractors. The
corresponding distributions for the unstable periodic orbits are investigated
for comparison. | nlin_CD |
Replication of Period-Doubling Route to Chaos in Coupled Systems with
Delay: In this study, replication of a period-doubling cascade in coupled systems
with delay is rigorously proved under certain assumptions, which guarantee the
existence of bounded solutions and replication of sensitivity. A novel
definition for replication of sensitivity is utilized, in which the proximity
of solutions is considered in an interval instead of a single point. Examples
with simulations supporting the theoretical results concerning sensitivity and
period-doubling cascade are provided. | nlin_CD |
Mean-field dynamics of a population of stochastic map neurons: We analyze the emergent regimes and the stimulus-response relationship of a
population of noisy map neurons by means of a mean-field model, derived within
the framework of cumulant approach complemented by the Gaussian closure
hypothesis. It is demonstrated that the mean-field model can qualitatively
account for stability and bifurcations of the exact system, capturing all the
generic forms of collective behavior, including macroscopic excitability,
subthreshold oscillations, periodic or chaotic spiking and chaotic bursting
dynamics. Apart from qualitative analogies, we find a substantial quantitative
agreement between the exact and the approximate system, as reflected in
matching of the parameter domains admitting the different dynamical regimes, as
well as the characteristic properties of the associated time series. The
effective model is further shown to reproduce with sufficient accuracy the
phase response curves of the exact system and the assembly's response to
external stimulation of finite amplitude and duration. | nlin_CD |
Comment on ``Analysis of chaotic motion and its shape dependence in a
generalized piecewise linear map'': Rajagopalan and Sabir [nlin.CD/0104021 and Phys. Rev. E 63, 057201 (2001)]
recently discussed deterministic diffusion in a piecewise linear map using an
approach developed by Fujisaka et al. We first show that they rederived the
random walk formula for the diffusion coefficient, which is known to be the
exact result for maps of Bernoulli type since the work of Fujisaka and
Grossmann [Z. Physik B {\bf 48}, 261 (1982)]. However, this correct solution is
at variance to the diffusion coefficient curve presented in their paper.
Referring to another existing approach based on Markov partitions, we answer
the question posed by the authors regarding solutions for more general
parameter values by recalling the finding of a fractal diffusion coefficient.
We finally argue that their model is not suitable for studying intermittent
behavior, in contrast to what was suggested in their paper. | nlin_CD |
Emergence and combinatorial accumulation of jittering regimes in spiking
oscillators with delayed feedback: Interaction via pulses is common in many natural systems, especially
neuronal. In this article we study one of the simplest possible systems with
pulse interaction: a phase oscillator with delayed pulsatile feedback. When the
oscillator reaches a specific state, it emits a pulse, which returns after
propagating through a delay line. The impact of an incoming pulse is described
by the oscillator's phase reset curve (PRC). In such a system we discover an
unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic
regular spiking solution bifurcates with several multipliers crossing the unit
circle at the same parameter value. The number of such critical multipliers
increases linearly with the delay and thus may be arbitrary large. This
bifurcation is accompanied by the emergence of numerous "jittering" regimes
with non-equal interspike intervals (ISIs). Each of these regimes corresponds
to a periodic solution of the system with a period roughly proportional to the
delay. The number of different "jittering" solutions emerging at the
bifurcation point increases exponentially with the delay. We describe the
combinatorial mechanism that underlies the emergence of such a variety of
solutions. In particular, we show how a periodic solution exhibiting several
distinct ISIs can imply the existence of multiple other solutions obtained by
rearranging of these ISIs. We show that the theoretical results for phase
oscillators accurately predict the behavior of an experimentally implemented
electronic oscillator with pulsatile feedback. | nlin_CD |
Fluctuation-dissipation relationship in chaotic dynamics: We consider a general N-degree-of-freedom dissipative system which admits of
chaotic behaviour. Based on a Fokker-Planck description associated with the
dynamics we establish that the drift and the diffusion coefficients can be
related through a set of stochastic parameters which characterize the steady
state of the dynamical system in a way similar to fluctuation-dissipation
relation in non-equilibrium statistical mechanics. The proposed relationship is
verified by numerical experiments on a driven double well system. | nlin_CD |
Invariant higher-order variational problems: We investigate higher-order geometric $k$-splines for template matching on
Lie groups. This is motivated by the need to apply diffeomorphic template
matching to a series of images, e.g., in longitudinal studies of Computational
Anatomy. Our approach formulates Euler-Poincar\'e theory in higher-order
tangent spaces on Lie groups. In particular, we develop the Euler-Poincar\'e
formalism for higher-order variational problems that are invariant under Lie
group transformations. The theory is then applied to higher-order template
matching and the corresponding curves on the Lie group of transformations are
shown to satisfy higher-order Euler-Poincar\'{e} equations. The example of
SO(3) for template matching on the sphere is presented explicitly. Various
cotangent bundle momentum maps emerge naturally that help organize the
formulas. We also present Hamiltonian and Hamilton-Ostrogradsky Lie-Poisson
formulations of the higher-order Euler-Poincar\'e theory for applications on
the Hamiltonian side. | nlin_CD |
On a liquid drop "falling" inside a heavier miscible fluid: We report a new type of drop instability, where the density difference
between the drop and the solvent is negative. We show that the drop falls
inside the solvent down to a minimum height, then fragmentation takes place and
secondary droplets rise up to the surface. We have developed a theoretical
model that captures the essential of the phenomenon and predicts the correct
scalings for the rise-up time and the minimum height. | nlin_CD |
Synchronization of coupled metronomes on two layers: Coupled metronomes serve as a paradigmatic model for exploring the collective
behaviors of complex dynamical systems, as well as a classical setup for
classroom demonstrations of synchronization phenomena. Whereas previous studies
of metronome synchronization have been concentrating on symmetric coupling
schemes, here we consider the asymmetric case by adopting the scheme of layered
metronomes. Specifically, we place two metronomes on each layer, and couple two
layers by placing one on top of the other. By varying the initial conditions of
the metronomes and adjusting the friction between the two layers, a variety of
synchronous patterns are observed in experiment, including the splay
synchronization (SS) state, the generalized splay synchronization (GSS) state ,
the anti-phase synchronization (APS) state, the in-phase delay synchronization
(IPDS) state, and the in-phase synchronization (IPS) state. In particular, the
IPDS state, in which the metronomes on each layer are synchronized in phase but
are of a constant phase delay to metronomes on the other layer, is observed for
the first time. In addition, a new technique based on audio signals is proposed
for pattern detection, which is more convenient and easier to apply than the
existing acquisition techniques. Furthermore, a theoretical model is developed
to explain the experimental observations, and is employed to explore the
dynamical properties of the patterns, including the basin distributions and the
pattern transitions. Our study sheds new lights on the collective behaviors of
coupled metronomes, and the developed setup can be used in the classroom for
demonstration purposes. | nlin_CD |
Structural, Dynamical and Symbolic Observability: From Dynamical Systems
to Networks: The concept of observability of linear systems initiated with Kalman in the
mid 1950s. Roughly a decade later, the observability of nonlinear systems
appeared. By such definitions a system is either observable or not. Continuous
measures of observability for linear systems were proposed in the 1970s and two
decades ago were adapted to deal with nonlinear dynamical systems. Related
topics developed either independently or as a consequence of these.
Observability has been recognized as an important feature to study complex
networks, but as for dynamical systems in the beginning the focus has been on
determining conditions for a network to be observable. In this relatively new
field previous and new results on observability merge either producing new
terminology or using terms, with well established meaning in other fields, to
refer to new concepts. Motivated by the fact that twenty years have passed
since some of these concepts were introduced in the field of nonlinear
dynamics, in this paper (i)~various aspects of observability will be reviewed,
and (ii)~it will be discussed in which ways networks could be ranked in terms
of observability. The aim is to make a clear distinction between concepts and
to understand what does each one contribute to the analysis and monitoring of
networks. Some of the main ideas are illustrated with simulations. | nlin_CD |
Mixed-lag synchronization in coupled counter-rotating oscillators: We report mixed lag synchronization in coupled counter-rotating oscillators.
The trajectories of counter-rotating oscillators has opposite directions of
rotation in uncoupled state. Under diffusive coupling via a scalar variable, a
mixed lag synchronization emerges when a parameter mismatch is induced in two
counter-rotating oscillators. In the state of mixed lag synchronization, one
pair of state variables achieve synchronization shifted in time while another
pair of state variables are in antisynchronization, however, they are too
shifted by the same time. Numerical example of the paradigmatic R{\"o}ssler
oscillator is presented and supported by electronic experiment. | nlin_CD |
Dynamics of a charged particle in a dissipative Fermi-Ulam model: The dynamics of a metallic particle confined between charged walls is
studied. One wall is fixed and the other moves smoothly and periodically in
time. Dissipation is considered by assuming a friction produced by the contact
between the particle and a rough surface. We investigate the phase space of the
simplified and complete versions of the model. Our results include (i)
coexistence of islands of regular motion with an attractor located at the low
energy portion of phase space in the complete model; and (ii) coexistence of
attractors with trajectories that present unlimited energy growth in the
simplified model. | nlin_CD |
Phenomena of complex analytic dynamics in the non-autonomous, nonlinear
ring system: The model system manifesting phenomena peculiar to complex analytic maps is
offered. The system is a non-autonomous ring cavity with nonlinear elements and
filters, | nlin_CD |
Fractal in the statistics of Goldbach partition: Some interesting chaos phenomena have been found in the difference of prime
numbers. Here we discuss a theme about the sum of two prime numbers, Goldbach
conjecture. This conjecture states that any even number could be expressed as
the sum of two prime numbers. Goldbach partition r(n) is the number of
representations of an even number n as the sum of two primes. This paper
analyzes the statistics of series r(n) (n=4,6,8,...). The familiar 3 period
oscillations in histogram of difference of consecutive primes appear in r(n).We
also find r(n) series could be divided into different levels period oscillation
series. The series in the same or different levels are all very similar, which
presents the obvious fractal phenomenon. Moreover, symmetry between the
statistics figure of sum and difference of two prime numbers are also
described. We find the estimate of Hardy-Littlewood could precisely depict
these phenomena. A rough analyzing for periodic behavior of r(n) is given by
symbolic dynamics theory at last. | nlin_CD |
Homoclinic orbits, and self-excited and hidden attractors in a
Lorenz-like system describing convective fluid motion: In this tutorial, we discuss self-excited and hidden attractors for systems
of differential equations. We considered the example of a Lorenz-like system
derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to
demonstrate the analysis of self-excited and hidden attractors and their
characteristics. We applied the fishing principle to demonstrate the existence
of a homoclinic orbit, proved the dissipativity and completeness of the system,
and found absorbing and positively invariant sets. We have shown that this
system has a self-excited attractor and a hidden attractor for certain
parameters. The upper estimates of the Lyapunov dimension of self-excited and
hidden attractors were obtained analytically. | nlin_CD |
Synchronization of Coupled Anizochronous Auto-Oscillating Systems: The particular properties of synchronization are discussed for coupled
auto-oscillating systems, which are characterized by non-quadratic law of
potential dependence on the coordinate. In particular, structure of the
parameter plane (frequency mismatch - coupling value) is considered for coupled
van der Pol - Duffing oscillators. The arrangement of synchronization tongues
and the particular properties of their internal structure in the parameter
space are revealed. The features of attractors in the phase space are
discussed. | nlin_CD |
From chaos of lines to Lagrangian structures in flux conservative fields: A numerical method is proposed in order to track field lines of
three-dimensional divergence free fields. Field lines are computed by a locally
valid Hamiltonian mapping, which is computed using a symplectic scheme. The
method is theoretically valid everywhere but at points where the field is null
or infinite. For any three dimensional flux conservative field for which
problematic points are sufficiently sparse, a systematic procedure is proposed
and implemented. Construction of field lines is achieved by means of tracers
and the introduction of various Hamiltonians adapted to the "geometrical state"
each line or tracer is. The states are artificially defined by an a priori
given frame of reference and Cartesian coordinates, and refer to a Hamiltonian
which is locally valid at the time step to be computed. This procedure ensures
the preservation of the volume (flux condition) during the iteration. This
method is first tested with an ABC-type flow. Its benefits when compared to
typical Runge-Kutta scheme are demonstrated. Potential use of the method to
exhibit "coherent" Lagrangian structures in a chaotic setting is shown. An
illustration to the computation of magnetic field lines resulting from a
three-dimensional MHD simulation is also provided. | nlin_CD |
Anomalous diffusion in a random nonlinear oscillator due to high
frequencies of the noise: We study the long time behaviour of a nonlinear oscillator subject to a
random multiplicative noise with a spectral density (or power-spectrum) that
decays as a power law at high frequencies. When the dissipation is negligible,
physical observables, such as the amplitude, the velocity and the energy of the
oscillator grow as power-laws with time. We calculate the associated scaling
exponents and we show that their values depend on the asymptotic behaviour of
the external potential and on the high frequencies of the noise. Our results
are generalized to include dissipative effects and additive noise. | nlin_CD |
On the concept of switching nonlinearity (a comment on "Switching
control of linear systems for generating chaos" by X. Liu, K-L. Teo, H. Zhang
and G. Chen): It is explained and stressed that the chaotic states in [1] are obtained by
means of nonlinear switching. | nlin_CD |
Transition from homogeneous to inhomogeneous steady states in
oscillators under cyclic coupling: We report a transition from homogeneous steady state to inhomogeneous steady
state in coupled oscillators, both limit cycle and chaotic, under cyclic
coupling and diffusive coupling as well when an asymmetry is introduced in
terms of a negative parameter mismatch. Such a transition appears in limit
cycle systems via pitchfork bifurcation as usual. Especially, when we focus on
chaotic systems, the transition follows a transcritical bifurcation for cyclic
coupling while it is a pitchfork bifurcation for the conventional diffusive
coupling. We use the paradigmatic Van der Pol oscillator as the limit cycle
system and a Sprott system as a chaotic system. We verified our results
analytically for cyclic coupling and numerically check all results including
diffusive coupling for both the limit cycle and chaotic systems. | nlin_CD |
Bouncing trimer: a random self-propelled particle, chaos and periodical
motions: A trimer is an object composed of three centimetrical stainless steel beads
equally distant and is predestined to show richer behaviours than the bouncing
ball or the bouncing dimer. The rigid trimer has been placed on a plate of a
electromagnetic shaker and has been vertically vibrated according to a
sinusoidal signal. The horizontal translational and rotational motions of the
trimer have been recorded for a range of frequencies between 25 and 100 Hz
while the amplitude of the forcing vibration was tuned for obtaining maximal
acceleration of the plate up to 10 times the gravity. Several modes have been
detected like e.g. rotational and pure translational motions. These modes are
found at determined accelerations of the plate and do not depend on the
frequency. By recording the time delays between two successive contacts when
the frequency and the amplitude are fixed, a mapping of the bouncing regime has
been constructed and compared to that of the dimer and the bouncing ball.
Period-2 and period-3 orbits have been experimentally observed. In these modes,
according to observations, the contact between the trimer and the plate is
persistent between two successive jumps. This persistence erases the memory of
the jump preceding the contact. A model is proposed and allows to explain the
values of the particular accelerations for which period-2 and period-3 modes
are observed. Finally, numerical simulations allow to reproduce the
experimental results. That allows to conclude that the friction between the
beads and the plate is the major dissipative process. | nlin_CD |
Transport properties of heavy particles in high Reynolds number
turbulence: The statistical properties of heavy particle trajectories in high Reynolds
numbers turbulent flows are analyzed. Dimensional analysis assuming Kolmogorov
scaling is compared with the result of numerical simulation using a synthetic
turbulence advecting field. The non-Markovian nature of the fluid velocity
statistics along the solid particle trajectories is put into evidence, and its
relevance in the derivation of Lagrangian transport models is discussed. | nlin_CD |
Algebraic decay in hierarchical graphs: We study the algebraic decay of the survival probability in open hierarchical
graphs. We present a model of a persistent random walk on a hierarchical graph
and study the spectral properties of the Frobenius-Perron operator. Using a
perturbative scheme, we derive the exponent of the classical algebraic decay in
terms of two parameters of the model. One parameter defines the geometrical
relation between the length scales on the graph, and the other relates to the
probabilities for the random walker to go from one level of the hierarchy to
another. The scattering resonances of the corresponding hierarchical quantum
graphs are also studied. The width distribution shows the scaling behavior
$P(\Gamma) \sim 1/\Gamma$. | nlin_CD |
Stability of Synchronized Chaos in Coupled Dynamical Systems: We consider the stability of synchronized chaos in coupled map lattices and
in coupled ordinary differential equations. Applying the theory of Hermitian
and positive semidefinite matrices we prove two results that give simple bounds
on coupling strengths which ensure the stability of synchronized chaos.
Previous results in this area involving particular coupling schemes (e.g.
global coupling and nearest neighbor diffusive coupling) are included as
special cases of the present work. | nlin_CD |
Iterated maps for clarinet-like systems: The dynamical equations of clarinet-like systems are known to be reducible to
a non-linear iterated map within reasonable approximations. This leads to time
oscillations that are represented by square signals, analogous to the Raman
regime for string instruments. In this article, we study in more detail the
properties of the corresponding non-linear iterations, with emphasis on the
geometrical constructions that can be used to classify the various solutions
(for instance with or without reed beating) as well as on the periodicity
windows that occur within the chaotic region. In particular, we find a regime
where period tripling occurs and examine the conditions for intermittency. We
also show that, while the direct observation of the iteration function does not
reveal much on the oscillation regime of the instrument, the graph of the high
order iterates directly gives visible information on the oscillation regime
(characterization of the number of period doubligs, chaotic behaviour, etc.). | nlin_CD |
Stepwise structure of Lyapunov spectra for many particle systems by a
random matrix dynamics: The structure of Lyapunov spectra for many particle systems with a random
interaction between the particles is discussed. The dynamics of the tangent
space is expressed as a master equation, which leads to a formula that connects
the positive Lyapunov exponents and the time correlations of the particle
interaction matrix. Applying this formula to one and two dimensional models we
investigate the stepwise structure of the Lyapunov spectra, which appear in the
region of small positive Lyapunov exponents. Long range interactions lead to a
clear separation of the Lyapunov spectra into a part exhibiting stepwise
structure and a part changing smoothly. The part of the Lyapunov spectrum
containing the stepwise structure is clearly distinguished by a wave like
structure in the eigenstates of the particle interaction matrix. The two
dimensional model has the same step widths as found numerically in a
deterministic chaotic system of many hard disks. | nlin_CD |
Local Analysis of Dissipative Dynamical Systems: Linear transformation techniques such as singular value decomposition (SVD)
have been used widely to gain insight into the qualitative dynamics of data
generated by dynamical systems. There have been several reports in the past
that had pointed out the susceptibility of linear transformation approaches in
the presence of nonlinear correlations. In this tutorial review, local
dispersion along with the surrogate testing is proposed to discriminate
nonlinear correlations arising in deterministic and non-deterministic settings. | nlin_CD |
Polynomial law for controlling the generation of n-scroll chaotic
attractors in an optoelectronic delayed oscillator: Controlled transitions between a hierarchy of n-scroll attractors are
investigated in a nonlinear optoelectronic oscillator. Using the system's
feedback strength as a control parameter, it is shown experimentally the
transition from Van der Pol-like attractors to 6-scroll, but in general, this
scheme can produce an arbitrary number of scrolls. The complexity of every
state is characterized by Lyapunov exponents and autocorrelation coefficients. | nlin_CD |
Spectral Statistics of Rectangular Billiards with Localized
Perturbations: The form factor $K(\tau)$ is calculated analytically to the order $\tau^3$ as
well as numerically for a rectangular billiard perturbed by a $\delta$-like
scatterer with an angle independent diffraction constant, $D$. The cases where
the scatterer is at the center and at a typical position in the billiard are
studied. The analytical calculations are performed in the semiclassical
approximation combined with the geometrical theory of diffraction. Non diagonal
contributions are crucial and are therefore taken into account.
The numerical calculations are performed for a self adjoint extension of a
$\delta$ function potential.
We calculate the angle dependent diffraction constant for an arbitrary
perturbing potential $U({\bf r})$, that is large in a finite but small region
(compared to the wavelength of the particles that in turn is small compared to
the size of the billiard). The relation to the idealized model of the
$\delta$-like scatterer is formulated. The angle dependent diffraction constant
is used for the analytic calculation of the form factor to the order $\tau^2$.
If the scatterer is at a typical position, the form factor is found to reduce
(in this order) to the one found for angle independent diffraction. If the
scatterer is at the center, the large degeneracy in the lengths of the orbits
involved leads to an additional small contribution to the form factor,
resulting of the angle dependence of the diffraction constant. The robustness
of the results is discussed. | nlin_CD |
A Generalization of Chaitin's Halting Probability Ωand Halting
Self-Similar Sets: We generalize the concept of randomness in an infinite binary sequence in
order to characterize the degree of randomness by a real number D>0. Chaitin's
halting probability \Omega is generalized to \Omega^D whose degree of
randomness is precisely D. On the basis of this generalization, we consider the
degree of randomness of each point in Euclidean space through its base-two
expansion. It is then shown that the maximum value of such a degree of
randomness provides the Hausdorff dimension of a self-similar set that is
computable in a certain sense. The class of such self-similar sets includes
familiar fractal sets such as the Cantor set, von Koch curve, and Sierpinski
gasket. Knowledge of the property of \Omega^D allows us to show that the
self-similar subset of [0,1] defined by the halting set of a universal
algorithm has a Hausdorff dimension of one. | nlin_CD |
The Lorenz system as a gradient-like system: We formulate, for continuous-time dynamical systems, a sufficient condition
to be a gradient-like system, i.e. that all bounded trajectories approach
stationary points and therefore that periodic orbits, chaotic attractors, etc.
do not exist. This condition is based upon the existence of an auxiliary
function defined over the state space of the system, in a way analogous to a
Lyapunov function for the stability of an equilibrium. For polynomial systems,
Lyapunov functions can be found computationally by using sum-of-squares
optimisation. We demonstrate this method by finding such an auxiliary function
for the Lorenz system. We are able to show that the system is gradient-like for
$0\leq\rho\leq12$ when $\sigma=10$ and $\beta=8/3$, significantly extending
previous results. The results are rigorously validated by a novel procedure:
First, an approximate numerical solution is found using finite-precision
floating-point sum-of-squares optimisation. We then prove that there exists an
exact solution close to this using interval arithmetic. | nlin_CD |
Bounds on a singular attractor in Euler using vorticity moments: A new rescaling of the vorticity moments and their growth terms is used to
characterise the evolution of anti-parallel vortices governed by the 3D Euler
equations. To suppress unphysical instabilities, the initial condition uses a
balanced profile for the initial magnitude of vorticity along with a new
algorithm for the initial vorticity direction. The new analysis uses a new
adaptation to the Euler equations of a rescaling of the vorticity moments
developed for Navier-Stokes analysis. All rescaled moments grow in time, with
the lower-order moments bounding the higher-order moments from above,
consistent with new results from several Navier-Stokes
calculations.Furthermore, if, as an inviscid flow evolves, this ordering is
assumed to hold, then a singular upper bound on the growth of these moments can
be used to provide a prediction of power law growth to compare against. There
is a significant period where the growth of the highest moments converges to
these singular bounds, demonstrating a tie between the strongest nonlinear
growth and how the rescaled vorticity moments are ordered. The logarithmic
growth of all the moments are calculated directly and the estimated singular
times for the different $D_m$ converge to a common value for the simulation in
the best domain. | nlin_CD |
Bogdanov-Takens resonance in time-delayed systems: We analyze the oscillatory dynamics of a time-delayed dynamical system
subjected to a periodic external forcing. We show that, for certain values of
the delay, the response can be greatly enhanced by a very small forcing
amplitude. This phenomenon is related to the presence of a Bogdanov- Takens
bifurcation and displays some analogies to other resonance phenomena, but also
substantial differences. | nlin_CD |
Local Extrema in Quantum Chaos: We numerically investigate the distribution of extrema of 'chaotic' Laplacian
eigenfunctions on two-dimensional manifolds. Our contribution is two-fold: (a)
we count extrema on grid graphs with a small number of randomly added edges and
show the behavior to coincide with the 1957 prediction of Longuet-Higgins for
the continuous case and (b) compute the regularity of their spatial
distribution using \textit{discrepancy}, which is a classical measure from the
theory of Monte Carlo integration. The first part suggests that grid graphs
with randomly added edges should behave like two-dimensional surfaces with
ergodic geodesic flow; in the second part we show that the extrema are more
regularly distributed in space than the grid $\mathbb{Z}^2$. | nlin_CD |
Emergence of a common generalized synchronization manifold in network
motifs of structurally different time-delay systems: We point out the existence of a transition from partial to global generalized
synchronization (GS) in symmetrically coupled structurally different time-delay
systems of different orders using the auxiliary system approach and the mutual
false nearest neighbor method. The present authors have recently reported that
there exists a common GS manifold even in an ensemble of structurally
nonidentical scalar time-delay systems with different fractal dimensions and
shown that GS occurs simultaneously with phase synchronization (PS). In this
paper we confirm that the above result is not confined just to scalar
one-dimensional time-delay systems alone but there exists a similar type of
transition even in the case of time-delay systems with different orders. We
calculate the maximal transverse Lyapunov exponent to evaluate the asymptotic
stability of the complete synchronization manifold of each of the main and the
corresponding auxiliary systems, which in turn ensures the stability of the GS
manifold between the main systems. Further we estimate the correlation
coefficient and the correlation of probability of recurrence to establish the
relation between GS and PS. We also calculate the mutual false nearest neighbor
parameter which doubly confirms the occurrence of the global GS manifold. | nlin_CD |
Soliton Propagation through a Disordered System: Statistics of the
Transmission Delay: We have studied the soliton propagation through a segment containing random
point-like scatterers. In the limit of small concentration of scatterers when
the mean distance between the scatterers is larger than the soliton width, a
method has been developed for obtaining the statistical characteristics of the
soliton transmission through the segment. The method is applicable for any
classical particle transferring through a disordered segment with the given
velocity transformation after each act of scattering. In the case of weak
scattering and relatively short disordered segment, the transmission time delay
of a fast soliton is mostly determined by the shifts of the soliton center
after each act of scattering. For sufficiently long segments the main
contribution to the delay is due to the shifts of the amplitude and velocity of
a fast soliton after each scatterer. Corresponding crossover lengths for both
cases of light and heavy solitons have been obtained. We have also calculated
the exact probability density function of the soliton transmission time delay
for a sufficiently long segment. In the case of weak identical scatterers it is
a universal function which depends on a sole parameter - mean number of
scatterers in a segment. | nlin_CD |
An Extension of Discrete Lagrangian Descriptors for Unbounded Maps: In this paper we provide an extension for the method of Discrete Lagrangian
Descriptors with the purpose of exploring the phase space of unbounded maps.
The key idea is to construct a working definition, that builds on the original
approach introduced in Lopesino et al. (2015), and which relies on stopping the
iteration of initial conditions when their orbits leave a certain region in the
plane. This criterion is partly inspired by the classical analysis used in
Dynamical Systems Theory to study the dynamics of maps by means of escape time
plots. We illustrate the capability of this technique to reveal the geometrical
template of stable and unstable invariant manifolds in phase space, and also
the intricate structure of chaotic sets and strange attractors, by applying it
to unveil the phase space of a well-known discrete time system, the H\'enon
map. | nlin_CD |
Dynamics of Sawtooth Map: 1. New Numerical Results: Some results of numerical study of the canonical map with a sawtooth force
are given and discovered new unexpected dynamical effects are described. In
particular, it is shown that if the values of the system parameter K belong to
the countable set determined by Ovsyannikov's theorem, separatrices of primary
resonances are not splitted and chaotic layers are not formed. One more set of
values of the parameter related to the other family of nondestructed
separatrices of primary resonances was found. The mechanism explaining the
stability of the primary resonance separatrix in the critical regime is found
and described. First secondary resonances were studies and for them were found
the K-values at which their separatrices are not splitted also. New problems
and open questions occurred in this connections whose solution can facilitate
the further development of the nonlinear Hamiltonian systems theory are
presented. | nlin_CD |
Impact of lag information on network inference: Extracting useful information from data is a fundamental challenge across
disciplines as diverse as climate, neuroscience, genetics, and ecology. In the
era of ``big data'', data is ubiquitous, but appropriated methods are needed
for gaining reliable information from the data. In this work we consider a
complex system, composed by interacting units, and aim at inferring which
elements influence each other, directly from the observed data. The only
assumption about the structure of the system is that it can be modeled by a
network composed by a set of $N$ units connected with $L$ un-weighted and
un-directed links, however, the structure of the connections is not known. In
this situation the inference of the underlying network is usually done by using
interdependency measures, computed from the output signals of the units. We
show, using experimental data recorded from randomly coupled electronic
R{\"o}ssler chaotic oscillators, that the information of the lag times obtained
from bivariate cross-correlation analysis can be useful to gain information
about the real connectivity of the system. | nlin_CD |
Experimental study of imperfect phase synchronization in the forced
Lorenz system: In this work we demonstrate for an experimental system, that exhibits the
Lorenz butterfly attractor behavior, that perfect chaotic phase synchronization
cannot be achieved in systems with an unbounded distribution of intrinsic time
scales. Instead, imperfect phase synchronization is characterized by the
occurrence of phase slips, associated to epochs of time during which the
chaotic oscillator exhibits a slower time scale. Interestingly, during phase
slips the chaotic oscillator keeps in sync with the drive, but with a different
locking ratio. | nlin_CD |
Synchronization of chaotic systems: A microscopic description: The synchronization of coupled chaotic systems represents a fundamental
example of self organization and collective behavior. This well-studied
phenomenon is classically characterized in terms of macroscopic parameters,
such as Lyapunov exponents, that help predict the systems transitions into
globally organized states. However, the local, microscopic, description of this
emergent process continues to elude us. Here we show that at the microscopic
level, synchronization is captured through a gradual process of topological
adjustment in phase space, in which the strange attractors of the two coupled
systems continuously converge, taking similar form, until complete topological
synchronization ensues. We observe the local nucleation of topological
synchronization in specific regions of the systems attractor, providing early
signals of synchrony, that appear significantly before the onset of complete
synchronization. This local synchronization initiates at the regions of the
attractor characterized by lower expansion rates, in which the chaotic
trajectories are least sensitive to slight changes in initial conditions. Our
findings offer an alternative description of synchronization in chaotic
systems, exposing its local embryonic stages that are overlooked by the
currently established global analysis. Such local topological synchronization
enables the identification of configurations where prediction of the state of
one system is possible from measurements on that of the other, even in the
absence of global synchronization. | nlin_CD |
Network as a Complex System: Information Flow Analysis: A new approach for the analysis of information flow on a network is suggested
using protocol parameters encapsulated in the package headers as functions of
time. The minimal number of independent parameters for a complete description
of the information flow (phase space dimension of the information flow) is
found to be about 10 - 12. | nlin_CD |
Nonlinear argumental oscillators: Stability criterion and attractor's
capture probability: The behaviour of a space-modulated, so-called "argumental" oscillator is
studied, which is represented by a model having an even-parity space-modulating
function. Analytic expressions of a stability criterion and of discrete energy
levels are given. Using an integrating factor and a Van der Pol representation
in the (amplitude, phase) space, an approximate implicit closed-form of the
solution is given. The probability to enter a stable-oscillation regime from
given initial conditions is calculated in symbolic form. These results allow an
analytic approach to stability and bifurcations of the system. They also allow
an assessment of the risk of occurrence of sustained large-amplitude
oscillations, when the phenomenon is to be avoided, and an assessment of the
conditions to apply to obtain oscillations whenever the phenomenon is desired. | nlin_CD |
Experimental Synchronization of Spatiotemporal Chaos in Nonlinear Optics: We demonstrate that a unidirectional coupling between a pattern forming
system and its replica induces complete synchronization of the slave to the
master system onto a spatiotemporal chaotic state. | nlin_CD |
Exact hopping and collision times for two hard discs in a box: We study the molecular dynamics of two discs undergoing Newtonian
("inertial") dynamics, with elastic collisions in a rectangular box. Using a
mapping to a billiard model and a key result from ergodic theory, we obtain
exact, analytical expressions for the mean times between the following events:
hops, i.e.~horizontal or vertical interchanges of the particles; wall
collisions; and disc collisions. To do so, we calculate volumes and
cross-sectional areas in the four-dimensional configuration space. We compare
the analytical results against Monte Carlo and molecular dynamics simulations,
with excellent agreement. | nlin_CD |
Asymptotically stable phase synchronization revealed by autoregressive
circle maps: A new type of nonlinear time series analysis is introduced, based on phases,
which are defined as polar angles in spaces spanned by a finite number of
delayed coordinates. A canonical choice of the polar axis and a related
implicit estimation scheme for the potentially underlying auto-regressive
circle map (next phase map) guarantee the invertibility of reconstructed phase
space trajectories to the original coordinates. The resulting Fourier
approximated, Invertibility enforcing Phase Space map (FIPS map) is well suited
to detect conditional asymptotic stability of coupled phases. This rather
general synchronization criterion unites two existing generalisations of the
old concept and can successfully be applied e.g. to phases obtained from ECG
and airflow recordings characterizing cardio-respiratory interaction. | nlin_CD |
Active vs passive scalar turbulence: Active and passive scalars transported by an incompressible two-dimensional
conductive fluid are investigated. It is shown that a passive scalar displays a
direct cascade towards the small scales while the active magnetic potential
builds up large-scale structures in an inverse cascade process. Correlations
between scalar input and particle trajectories are found to be responsible for
those dramatic differences as well as for the behavior of dissipative
anomalies. | nlin_CD |
On the semiclassical expansion for 1-dim $x^N$ potentials: In the present paper we study the structure of the WKB series for the
polynomial potential $V(x)=x^N$ ($N$ even). In particular, we obtain relatively
simple recurrence formula of the coefficients $\s'_k$ of the semiclassical
approximation and of the WKB terms for the energy eigenvalues. | nlin_CD |
Time-Delayed Feedback Control Design Beyond the Odd Number Limitation: We present an algorithm for a time-delayed feedback control design to
stabilize periodic orbits with an odd number of positive Floquet exponents in
autonomous systems. Due to the so-called odd number theorem such orbits have
been considered as uncontrollable by time-delayed feedback methods. However,
this theorem has been refuted by a counterexample and recently a corrected
version of the theorem has been proved. In our algorithm, the control matrix is
designed using a relationship between Floquet multipliers of the systems
controlled by time-delayed and proportional feedback. The efficacy of the
algorithm is demonstrated with the Lorenz and Chua systems. | nlin_CD |
Equivariant singularity analysis of the 2:2 resonance: We present a general analysis of the bifurcation sequences of 2:2 resonant
reversible Hamiltonian systems invariant under spatial $\Z_2\times\Z_2$
symmetry. The rich structure of these systems is investigated by a singularity
theory approach based on the construction of a universal deformation of the
detuned Birkhoff normal form. The thresholds for the bifurcations are computed
as asymptotic series also in terms of physical quantities for the original
system. | nlin_CD |
Statistics of the inverse-cascade regime in two-dimensional
magnetohydrodynamic turbulence: We present a detailed direct numerical simulation of statistically steady,
homogeneous, isotropic, two-dimensional magnetohydrodynamic (2D MHD)
turbulence. Our study concentrates on the inverse cascade of the magnetic
vector potential. We examine the dependence of the statistical properties of
such turbulence on dissipation and friction coefficients. We extend earlier
work sig- nificantly by calculating fluid and magnetic spectra, probability
distribution functions (PDFs) of the velocity, magnetic, vorticity, current,
stream-function, and magnetic-vector-potential fields and their increments. We
quantify the deviations of these PDFs from Gaussian ones by computing their
flatnesses and hyperflatnesses. We also present PDFs of the Okubo-Weiss
parameter, which distin- guishes between vortical and extensional flow regions,
and its magnetic analog. We show that the hyperflatnesses of PDFs of the
increments of the stream-function and the magnetic vector potential exhibit
significant scale dependence and we examine the implication of this for the
multiscaling of structure functions. We compare our results with those of
earlier studies. | nlin_CD |
Dynamics of delayed-coupled chaotic logistic maps: influence of network
topology, connectivity and delay times: We review our recent work on the synchronization of a network of
delay-coupled maps, focusing on the interplay of the network topology and the
delay times that take into account the finite velocity of propagation of
interactions. We assume that the elements of the network are identical ($N$
logistic maps in the regime where the individual maps, without coupling, evolve
in a chaotic orbit) and that the coupling strengths are uniform throughout the
network. We show that if the delay times are sufficiently heterogeneous, for
adequate coupling strength the network synchronizes in a spatially homogeneous
steady-state, which is unstable for the individual maps without coupling. This
synchronization behavior is referred to as ``suppression of chaos by random
delays'' and is in contrast with the synchronization when all the interaction
delay times are homogeneous, because with homogeneous delays the network
synchronizes in a state where the elements display in-phase time-periodic or
chaotic oscillations. We analyze the influence of the network topology
considering four different types of networks: two regular (a ring-type and a
ring-type with a central node) and two random (free-scale Barabasi-Albert and
small-world Newman-Watts). We find that when the delay times are sufficiently
heterogeneous the synchronization behavior is largely independent of the
network topology but depends on the networks connectivity, i.e., on the average
number of neighbors per node. | nlin_CD |
The statistical properties of the city transport in Cuernavaca (Mexico)
and Random matrix ensembles: We analyze statistical properties of the city bus transport in Cuernavaca
(Mexico) and show that the bus arrivals display probability distributions
conforming those given by the Unitary Ensemble of random matrices. | nlin_CD |
Wave turbulence buildup in a vibrating plate: We report experimental and numerical results on the buildup of the energy
spectrum in wave turbulence of a vibrating thin elastic plate. Three steps are
observed: first a short linear stage, then the turbulent spectrum is
constructed by the propagation of a front in wave number space and finally a
long time saturation due to the action of dissipation. The propagation of a
front at the second step is compatible with scaling predictions from the Weak
Turbulence Theory. | nlin_CD |
Exponentially growing solutions in homogeneous Rayleigh-Benard
convection: It is shown that homogeneous Rayleigh-Benard flow, i.e., Rayleigh-Benard
turbulence with periodic boundary conditions in all directions and a volume
forcing of the temperature field by a mean gradient, has a family of exact,
exponentially growing, separable solutions of the full non-linear system of
equations. These solutions are clearly manifest in numerical simulations above
a computable critical value of the Rayleigh number. In our numerical
simulations they are subject to secondary numerical noise and resolution
dependent instabilities that limit their growth to produce statistically steady
turbulent transport. | nlin_CD |
Colored noise induces synchronization of limit cycle oscillators: Driven by various kinds of noise, ensembles of limit cycle oscillators can
synchronize. In this letter, we propose a general formulation of
synchronization of the oscillator ensembles driven by common colored noise with
an arbitrary power spectrum. To explore statistical properties of such colored
noise-induced synchronization, we derive the stationary distribution of the
phase difference between two oscillators in the ensemble. This analytical
result theoretically predicts various synchronized and clustered states induced
by colored noise and also clarifies that these phenomena have a different
synchronization mechanism from the case of white noise. | nlin_CD |
Multistability and transition to chaos in the degenerate Hamiltonian
system with weak nonlinear dissipative perturbation: The effect of small nonlinear dissipation on the dynamics of system with
stochastic web which is linear oscillator driven by pulses is studied. The
scenario of coexisting attractors evolution with the increase of nonlinear
dissipation is revealed. It is shown that the period-doubling transition to
chaos is possible only for third order resonance and only hard transitions can
be seen for all other resonances. | nlin_CD |
Sustained turbulence in the three-dimensional Gross-Pitaevskii model: We study the 3D forced-dissipated Gross-Pitaevskii equation. We force at
relatively low wave numbers, expecting to observe a direct energy cascade and a
consequent power-law spectrum of the form $k^{-\alpha}$. Our numerical results
show that the exponent $\alpha$ strongly depends on how the inverse particle
cascade is attenuated at $k$'s lower than the forcing wave number. If the
inverse cascade is arrested by a friction at low $k$'s, we observe an exponent
which is in good agreement with the weak wave turbulence prediction $k^{-1}$.
For a hypo-viscosity, a $k^{-2}$ spectrum is observed which we explain using a
critical balance argument. In simulations without any low-$k$ dissipation, a
condensate at $k=0$ is growing and the system goes through a strongly-turbulent
transition from a four-wave to a three-wave weak turbulence acoustic regime
with $k^{-3/2}$ Zakharov-Sagdeev spectrum. In this regime, we also observe a
spectrum for the incompressible kinetic energy which formally resembles the
Kolmogorov $k^{-5/3}$, but whose correct explanation should be in terms of the
Kelvin wave turbulence. The probability density functions for the velocities
and the densities are also discussed. | nlin_CD |
Stabilizing unstable periodic orbits in the Lorenz equations using
time-delayed feedback control: For many years it was believed that an unstable periodic orbit with an odd
number of real Floquet multipliers greater than unity cannot be stabilized by
the time-delayed feedback control mechanism of Pyragus. A recent paper by
Fiedler et al uses the normal form of a subcritical Hopf bifurcation to give a
counterexample to this theorem. Using the Lorenz equations as an example, we
demonstrate that the stabilization mechanism identified by Fiedler et al for
the Hopf normal form can also apply to unstable periodic orbits created by
subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our
analysis focuses on a particular codimension-two bifurcation that captures the
stabilization mechanism in the Hopf normal form example, and we show that the
same codimension-two bifurcation is present in the Lorenz equations with
appropriately chosen Pyragus-type time-delayed feedback. This example suggests
a possible strategy for choosing the feedback gain matrix in Pyragus control of
unstable periodic orbits that arise from a subcritical Hopf bifurcation of a
stable equilibrium. In particular, our choice of feedback gain matrix is
informed by the Fiedler et al example, and it works over a broad range of
parameters, despite the fact that a center-manifold reduction of the
higher-dimensional problem does not lead to their model problem. | nlin_CD |
Synchronization of coupled anizochronous oscillators: The particular properties of synchronization are discussed for the system of
coupled van der Pol - Duffing oscillators. The arrangement of synchronization
tongues and the particular properties of their internal structure in the
parameter space are revealed. The features of attractors in the phase space and
in the Poincare section are considered. | nlin_CD |
Intra-layer synchronization in multiplex networks: We study synchronization of $N$ oscillators indirectly coupled through a
medium which is inhomogeneous and has its own dynamics. The system is
formalized in terms of a multilayer network, where the top layer is made of
disconnected oscillators and the bottom one, modeling the medium, consists of
oscillators coupled according to a given topology. The different dynamics of
the medium and the top layer is accounted by including a frequency mismatch
between them. We show a novel regime of synchronization as intra-layer
coherence does not necessarily require inter-layer coherence. This regime
appears under mild conditions on the bottom layer: arbitrary topologies may be
considered, provided that they support synchronization of the oscillators of
the medium. The existence of a density-dependent threshold as in quorum-sensing
phenomena is also demonstrated. | nlin_CD |
Anomalous diffusion in random dynamical systems: Consider a chaotic dynamical system generating Brownian motion-like
diffusion. Consider a second, non-chaotic system in which all particles
localize. Let a particle experience a random combination of both systems by
sampling between them in time. What type of diffusion is exhibited by this {\em
random dynamical system}? We show that the resulting dynamics can generate
anomalous diffusion, where in contrast to Brownian normal diffusion the mean
square displacement of an ensemble of particles increases nonlinearly in time.
Randomly mixing simple deterministic walks on the line we find anomalous
dynamics characterised by ageing, weak ergodicity breaking, breaking of
self-averaging and infinite invariant densities. This result holds for general
types of noise and for perturbing nonlinear dynamics in bifurcation scenarios. | nlin_CD |
Lyapunov Exponents, Transport and the Extensivity of Dimensional Loss: An explicit relation between the dimensional loss ($\Delta D$), entropy
production and transport is established under thermal gradients, relating the
microscopic and macroscopic behaviors of the system. The extensivity of $\Delta
D$ in systems with bulk behavior follows from the relation. The maximum
Lyapunov exponents in thermal equilibrium and $\Delta D$ in non-equilibrium
depend on the choice of heat-baths, while their product is unique and
macroscopic. Finite size corrections are also computed and all results are
verified numerically. | nlin_CD |
Chaos or Noise - Difficulties of a Distinction: In experiments, the dynamical behavior of systems is reflected in time
series. Due to the finiteness of the observational data set it is not possible
to reconstruct the invariant measure up to arbitrary fine resolution and
arbitrary high embedding dimension. These restrictions limit our ability to
distinguish between signals generated by different systems, such as regular,
chaotic or stochastic ones, when analyzed from a time series point of view. We
propose to classify the signal behavior, without referring to any specific
model, as stochastic or deterministic on a certain scale of the resolution
$\epsilon$, according to the dependence of the $(\epsilon,\tau)$-entropy,
$h(\epsilon, \tau)$, and of the finite size Lyapunov exponent,
$\lambda(\epsilon)$, on $\epsilon$. | nlin_CD |
A nonlinear system with weak dissipation under external force on
incommensurate frequency: coexistence of attractors: The behaviour of the 2D model system - the Ikeda map - is investigated in the
weakly dissipative regime under external forcing on the incommensurate
frequency. Coexistence of a large number of stable invariant curves is shown.
Dependence of the number of coexisting attractors on the external forcing
amplitude and nonlinearity parameter is investigated. | nlin_CD |
On one-dimensional Schroedinger problems allowing polynomial solutions: We discuss the explicit construction of the Schroedinger equations admitting
a representation through some family of general polynomials. Almost all
solvable quantum potentials are shown to be generated by this approach.
Some generalization has also been performed in higher-dimensional problems. | nlin_CD |
Coherent libration to coherent rotational dynamics via chimeralike
states and clustering in Josephson Junction array: An array of excitable Josephson junctions under global mean-field interaction
and a common periodic forcing shows emergence of two important classes of
coherent dynamics, librational and rotational motion in the weaker and stronger
coupling limits, respectively, with transitions to chimeralike states and
clustered states in the intermediate coupling range. In this numerical study,
we use the Kuramoto complex order parameter and introduce two measures, a
libration index and a clustering index to characterize the dynamical regimes
and their transition and locate them in a parameter plane. | nlin_CD |
Comparison of averages of flows and maps: It is shown that in transient chaos there is no direct relation between
averages in a continuos time dynamical system (flow) and averages using the
analogous discrete system defined by the corresponding Poincare map. In
contrast to permanent chaos, results obtained from the Poincare map can even be
qualitatively incorrect. The reason is that the return time between
intersections on the Poincare surface becomes relevant. However, after
introducing a true-time Poincare map, quantities known from the usual Poincare
map, such as conditionally invariant measure and natural measure, can be
generalized to this case. Escape rates and averages, e.g. Liapunov exponents
and drifts can be determined correctly using these novel measures. Significant
differences become evident when we compare with results obtained from the usual
Poincare map. | nlin_CD |
Stochastic approach to diffusion inside the chaotic layer of a resonance: We model chaotic diffusion, in a symplectic 4D map by using the result of a
theorem that was developed for stochastically perturbed integrable Hamiltonian
systems. We explicitly consider a map defined by a free rotator (FR) coupled to
a standard map (SM). We focus in the diffusion process in the action, $I$, of
the FR, obtaining a semi--numerical method to compute the diffusion
coefficient. We study two cases corresponding to a thick and a thin chaotic
layer in the SM phase space and we discuss a related conjecture stated in the
past. In the first case the numerically computed probability density function
for the action $I$ is well interpolated by the solution of a Fokker-Planck
(F-P) equation, whereas it presents a non--constant time delay respect to the
concomitant F-P solution in the second case suggesting the presence of an
anomalous diffusion time scale. The explicit calculation of a diffusion
coefficient for a 4D symplectic map can be useful to understand the slow
diffusion observed in Celestial Mechanics and Accelerator Physics. | nlin_CD |
Community consistency determines the stability transition window of
power-grid nodes: The synchrony of electric power systems is important in order to maintain
stable electricity supply. Recently, the measure basin stability was introduced
to quantify a node's ability to recover its synchronization when perturbed. In
this work, we focus on how basin stability depends on the coupling strength
between nodes. We use the Chilean power grid as a case study. In general, basin
stability goes from zero to one as coupling strength increases. However, this
transition does not happen at the same value for different nodes. By
understanding the transition for individual nodes, we can further characterize
their role in the power-transmission dynamics. We find that nodes with an
exceptionally large transition window also have a low community consistency. In
other words, they are hard to classify to one community when applying a
community detection algorithm. This also gives an efficient way to identify
nodes with a long transition window (which is computationally time consuming).
Finally, to corroborate these results, we present a stylized example network
with prescribed community structures that captures the mentioned
characteristics of basin stability transition and recreates our observations. | nlin_CD |
Data assimilation as a nonlinear dynamical systems problem: Stability
and convergence of the prediction-assimilation system: We study prediction-assimilation systems, which have become routine in
meteorology and oceanography and are rapidly spreading to other areas of the
geosciences and of continuum physics. The long-term, nonlinear stability of
such a system leads to the uniqueness of its sequentially estimated solutions
and is required for the convergence of these solutions to the system's true,
chaotic evolution. The key ideas of our approach are illustrated for a
linearized Lorenz system. Stability of two nonlinear prediction-assimilation
systems from dynamic meteorology is studied next via the complete spectrum of
their Lyapunov exponents; these two systems are governed by a large set of
ordinary and of partial differential equations, respectively. The degree of
data-induced stabilization is crucial for the performance of such a system.
This degree, in turn, depends on two key ingredients: (i) the observational
network, either fixed or data-adaptive; and (ii) the assimilation method. | nlin_CD |
Towards a geometrical classification of statistical conservation laws in
turbulent advection: The paper revisits the compressible Kraichnan model of turbulent advection in
order to derive explicit quantitative relations between scaling exponents and
Lagrangian particle configuration geometry. | nlin_CD |
Stability of Fixed Points in Generalized Fractional Maps of the Orders
$0< α<1$: Caputo fractional (with power-law kernels) and fractional (delta) difference
maps belong to a more widely defined class of generalized fractional maps,
which are discrete convolutions with some power-law-like functions. The
conditions of the asymptotic stability of the fixed points for maps of the
orders $0< \alpha <1$ that are derived in this paper are narrower than the
conditions of stability for the discrete convolution equations in general and
wider than the well-known conditions of stability for the fractional difference
maps. The derived stability conditions for the fractional standard and logistic
maps coincide with the results previously observed in numerical simulations. In
nonlinear maps, one of the derived limits of the fixed-point stability
coincides with the fixed-point - asymptotically period two bifurcation point. | nlin_CD |
Method of Asymptotics beyond All Orders and Restriction on Maps: The method of asymptotics beyond all orders (ABAO) is known to be a useful
tool to investigate separatrix splitting of several maps. For a class of
simplectic maps, the form of maps is shown to be restricted by the conditions
for the ABAO method to work well. More over, we check that the standard map,
H\' enon map and the cubic map satisfy the restrcitions. | nlin_CD |
Dispersive and friction-induced stabilization of an inverse cascade. The
theory for the Kolmogorov flow in the slightly supercritical regime: We discuss the stabilisation of the inverse cascade in the large scale
instability of the Kolmogorov flow described by the complete Cahn-Hilliard
equation with inclusion of $\beta$ effect, large-scale friction and deformation
radius. The friction and the $\beta$values halting the inverse cascade at the
various possible intermediate states are calculated by means of singular
perturbation techniques and compared to the values resulting from numerical
simulation of the complete Cahn-Hilliard equation. The excellent agreement
validates the theory. Our main result is that the critical values of friction
or $\beta$ halting the inverse cascade scale exponentially as a function of the
jet separation in the final flow, contrary to previous theories and
phenomenological approach. | nlin_CD |
Alternative determinism principle for topological analysis of chaos: The topological analysis of chaos based on a knot-theoretic characterization
of unstable periodic orbits has proved a powerful method, however knot theory
can only be applied to three-dimensional systems. Still, the core principles
upon which this approach is built, determinism and continuity, apply in any
dimension. We propose an alternative framework in which these principles are
enforced on triangulated surfaces rather than curves and show that in dimension
three our approach numerically predicts the correct topological entropies for
periodic orbits of the horseshoe map. | nlin_CD |
Translationally invariant cumulants in energy cascade models of
turbulence: In the context of random multiplicative energy cascade processes, we derive
analytical expressions for translationally invariant one- and two-point
cumulants in logarithmic field amplitudes. Such cumulants make it possible to
distinguish between hitherto equally successful cascade generator models and
hence supplement lowest-order multifractal scaling exponents and multiplier
distributions. | nlin_CD |
Singular fractal dimension at periodicity cascades in parameters spaces: In the parameter spaces of nonlinear dynamical systems, we investigate the
boundaries between periodicity and chaos and unveil the existence of fractal
sets characterized by a singular fractal dimension. This dimension stands out
from the typical fractal dimensions previously considered universal for these
parameter boundaries. We show that the singular fractal sets dwell along
parameter curves, called extreme curves, that intersect periodicity cascades at
their center of stability in all scales of parameters spaces. The results
reported here are generally demonstrated for the class of one-dimensional maps
with at least two control parameters, generalizations to other classes of
systems are possible. | nlin_CD |
Quantum Chaos of a particle in a square well : Competing Length Scales
and Dynamical Localization: The classical and quantum dynamics of a particle trapped in a one-dimensional
infinite square well with a time periodic pulsed field is investigated. This is
a two-parameter non-KAM generalization of the kicked rotor, which can be seen
as the standard map of particles subjected to both smooth and hard potentials.
The virtue of the generalization lies in the introduction of an extra parameter
R which is the ratio of two length scales, namely the well width and the field
wavelength. If R is a non-integer the dynamics is discontinuous and non-KAM. We
have explored the role of R in controlling the localization properties of the
eigenstates. In particular the connection between classical diffusion and
localization is found to generalize reasonably well. In unbounded chaotic
systems such as these, while the nearest neighbour spacing distribution of the
eigenvalues is less sensitive to the nature of the classical dynamics, the
distribution of participation ratios of the eigenstates proves to be a
sensitive measure; in the chaotic regimes the latter being lognormal. We find
that the tails of the well converged localized states are exponentially
localized despite the discontinuous dynamics while the bulk part shows
fluctuations that tend to be closer to Random Matrix Theory predictions. Time
evolving states show considerable R dependence and tuning R to enhance
classical diffusion can lead to significantly larger quantum diffusion for the
same field strengths, an effect that is potentially observable in present day
experiments. | nlin_CD |
Statistical properties of the continuum Salerno model: The statistical properties of the Salerno model is investigated.
In particular, a comparison between the coherent and partially coherent wave
modes is made for the case of a random phased wave packet. It is found that the
random phased induced spectral broadening gives rise to damping of
instabilities, but also a broadening of the instability region in
quasi-particle momentum space. The results can be of significance for
condensation of magnetic moment bosons in deep optical lattices. | nlin_CD |
Effect of Random Parameter Switching on Commensurate Fractional Order
Chaotic Systems: The paper explores the effect of random parameter switching in a fractional
order (FO) unified chaotic system which captures the dynamics of three popular
sub-classes of chaotic systems i.e. Lorenz, Lu and Chen's family of attractors.
The disappearance of chaos in such systems which rapidly switch from one family
to the other has been investigated here for the commensurate FO scenario. Our
simulation study show that a noise-like random variation in the key parameter
of the unified chaotic system along with a gradual decrease in the commensurate
FO is capable of suppressing the chaotic fluctuations much earlier than that
with the fixed parameter one. The chaotic time series produced by such random
parameter switching in nonlinear dynamical systems have been characterized
using the largest Lyapunov exponent (LLE) and Shannon entropy. The effect of
choosing different simulation techniques for random parameter FO switched
chaotic systems have also been explored through two frequency domain and three
time domain methods. Such a noise-like random switching mechanism could be
useful for stabilization and control of chaotic oscillation in many real-world
applications. | nlin_CD |
Implementation of PDE models of cardiac dynamics on GPUs using OpenCL: Graphical processing units (GPUs) promise to revolutionize scientific
computing in the near future. Already, they allow almost real-time integration
of simplified numerical models of cardiac tissue dynamics. However, the
integration methods that have been developed so far are typically of low order
and use single precision arithmetics. In this work, we describe numerical
implementation of double precision integrators required by, e.g., matrix-free
Newton-Krylov solvers and compare several higher order, fully explicit
numerical methods using finite-difference discretization of a range of models
of two-dimensional cardiac tissue. | nlin_CD |
Length-scale estimates for the LANS-alpha equations in terms of the
Reynolds number: Foias, Holm & Titi \cite{FHT2} have settled the problem of existence and
uniqueness for the 3D \lans equations on periodic box $[0,L]^{3}$. There still
remains the problem, first introduced by Doering and Foias \cite{DF} for the
Navier-Stokes equations, of obtaining estimates in terms of the Reynolds number
$\Rey$, whose character depends on the fluid response, as opposed to the
Grashof number, whose character depends on the forcing. $\Rey$ is defined as
$\Rey = U\ell/\nu$ where $U$ is a bounded spatio-temporally averaged
Navier-Stokes velocity field and $\ell$ the characteristic scale of the
forcing. It is found that the inverse Kolmogorov length is estimated by
$\ell\lambda_{k}^{-1} \leq c (\ell/\alpha)^{1/4}\Rey^{5/8}$. Moreover, the
estimate of Foias, Holm & Titi for the fractal dimension of the global
attractor, in terms of $\Rey$, comes out to be $$ d_{F}(\mathcal{A}) \leq c
\frac{V_{\alpha}V_{\ell}^{1/2}}{(L^{2}\lambda_{1})^{9/8}} \Rey^{9/4} $$ where
$V_{\alpha} = (L/(\ell\alpha)^{1/2})^{3}$ and $V_{\ell} = (L/\ell)^{3}$. It is
also shown that there exists a series of time-averaged inverse squared length
scales whose members, $\left<\kappa_{n,0}^2\right>$, %, are related to the
$2n$th-moments of the energy spectrum when $\alpha\to 0$. are estimated as
$(n\geq 1)$ $$ \ell^{2}\left<\kappa_{n,0}^2\right> \leq
c_{n,\alpha}V_{\alpha}^{\frac{n-1}{n}} \Rey^{{11/4} -
\frac{7}{4n}}(\ln\Rey)^{\frac{1}{n}} + c_{1}\Rey(\ln\Rey) . $$ The upper bound
on the first member of the hierarchy $\left<\kappa_{1,0}^2\right>$ coincides
with the inverse squared Taylor micro-scale to within log-corrections. | nlin_CD |
Chimera death induced by the mean-field diffusive coupling: Recently a novel dynamical state, called the {\it chimera death}, is
discovered in a network of non locally coupled identical oscillators [A.
Zakharova, M. Kapeller, and E. Sch\"oll, Phy.Rev.Lett. 112, 154101 (2014)],
which is defined as the coexistence of spatially coherent and incoherent
oscillation death state. This state arises due to the interplay of non locality
and symmetry breaking and thus bridges the gap between two important dynamical
states, namely the chimera and oscillation death. In this paper we show that
the chimera death can be induced in a network of generic identical oscillators
with mean-field diffusive coupling and thus we establish that a non local
coupling is not essential to obtain chimera death. We identify a new transition
route to the chimera death state, namely the transition from in-phase
synchronized oscillation to chimera death via global amplitude death state. We
ascribe the occurrence of chimera death to the bifurcation structure of the
network in the limiting condition and show that multi-cluster chimera death
states can be achieved by a proper choice of initial conditions. | nlin_CD |
Time Scaling of Chaotic Systems: Application to Secure Communications: The paper deals with time-scaling transformations of dynamical systems. Such
scaling functions operate a change of coordinates on the time axis of the
system trajectories preserving its phase portrait. Exploiting this property, a
chaos encryption technique to transmit a binary signal through an analog
channel is proposed. The scheme is based on a suitable time-scaling function
which plays the role of a private key. The encoded transmitted signal is proved
to resist known decryption attacks offering a secure and reliable
communication. | nlin_CD |
Generic Superweak Chaos Induced by Hall Effect: We introduce and study the "kicked Hall system" (KHS), i.e., charged
particles periodically kicked in the presence of uniform magnetic
($\mathbf{B}$) and electric ($\mathbf{E}$) fields that are perpendicular to
each other and to the kicking direction. We show that for resonant values of
$B$ and $E$ and in the weak-chaos regime of sufficiently small nonintegrability
parameter $\kappa$ (the kicking strength), there exists a \emph{generic} family
of periodic kicking potentials for which the Hall effect from $\mathbf{B}$ and
$\mathbf{E}$ significantly suppresses the weak chaos, replacing it by
\emph{"superweak"} chaos (SWC). This means that the system behaves as if the
kicking strength were $\kappa ^2$ rather than $\kappa$. For $E=0$, SWC is known
to be a classical fingerprint of quantum antiresonance but it occurs under much
less generic conditions, in particular only for very special kicking
potentials. Manifestations of SWC are a decrease in the instability of periodic
orbits and a narrowing of the chaotic layers, relative to the ordinary
weak-chaos case. Also, for global SWC, taking place on an infinite "stochastic
web" in phase space, the chaotic diffusion on the web is much slower than the
weak-chaos one. Thus, the Hall effect can be relatively stabilizing for small
$\kappa$. In some special cases, the effect is shown to cause ballistic motion
for almost all parameter values. The generic global SWC on stochastic webs in
the KHS appears to be the two-dimensional closest analog to the Arnol'd web in
higher dimensional systems. | nlin_CD |
Spectral Simplicity of Apparent Complexity, Part I: The
Nondiagonalizable Metadynamics of Prediction: Virtually all questions that one can ask about the behavioral and structural
complexity of a stochastic process reduce to a linear algebraic framing of a
time evolution governed by an appropriate hidden-Markov process generator. Each
type of question---correlation, predictability, predictive cost, observer
synchronization, and the like---induces a distinct generator class. Answers are
then functions of the class-appropriate transition dynamic. Unfortunately,
these dynamics are generically nonnormal, nondiagonalizable, singular, and so
on. Tractably analyzing these dynamics relies on adapting the recently
introduced meromorphic functional calculus, which specifies the spectral
decomposition of functions of nondiagonalizable linear operators, even when the
function poles and zeros coincide with the operator's spectrum. Along the way,
we establish special properties of the projection operators that demonstrate
how they capture the organization of subprocesses within a complex system.
Circumventing the spurious infinities of alternative calculi, this leads in the
sequel, Part II, to the first closed-form expressions for complexity measures,
couched either in terms of the Drazin inverse (negative-one power of a singular
operator) or the eigenvalues and projection operators of the appropriate
transition dynamic. | nlin_CD |
Separating the influence of Brain Signals from the Dynamics of Heart: ECG signals appear to be quite complex. In this paper, we present results,
which show that a normal ECG signal, which is a function of time can be
transformed into a relatively simpler signal by stretching the time in a
predetermined way. Before such a transformation, if you were to analyze various
packets of the data for the Trans-Spectral Coherence (TSC) you could confirm
that the signal indeed is very complicated. This is because the TSC gives us an
idea of how various harmonics in a spectrum are related to one another. The
coherence dramatically improved once we found an intermediate variable..
However, there was one hurdle. To find this variable, we needed to postulate
that the complexified data lies on a 4-sheeted Riemann-Surface. With this
insight, we could identify a proper time transformation which led to extremely
high TSC.
The transformed signal is quite simple. We can now rearrange the ECG data in
terms of a set of functions in an affine space, which we can explicitly
calculate from the data. This reveals that the dynamics of the heart, if freed
from the external influence is quite simple. | nlin_CD |
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