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arXiv1996

Period p-tuplings in coupled maps

We study the critical behavior (CB) of all period $p$-tuplings $(p \!=\!2,3,4,\dots)$ in $N$ $(N \!=\! 2,3,4,\dots)$ symmetrically coupled one-dimensional maps. We first investigate the CB for the $N=2$ case of two coupled maps, using a renormalization method. Three (five) kinds of fixed points of the renormalization transformation and their relevant ``coupling eigenvalues'' associated with coupling perturbations are found in the case of even (odd) $p$. We next study the CB for the linear- and nonlinear-coupling cases (a coupling is called linear or nonlinear according to its leading term), and confirm the renormalization results. Both the structure of the critical set (set of the critical points) and the CB vary according as the coupling is linear or nonlinear. Finally, the results of the two coupled maps are extended to many coupled maps with $N \geq 3$, in which the CB depends on the range of coupling.

chao-dyn↗
arXiv1996

Characterization of Landau-Zener Transitions in Systems with Complex Spectra

This paper is concerned with the study of one-body dissipation effects in idealized models resembling a nucleus. In particular, we study the quantum mechanics of a free particle that collides elastically with the slowly moving walls of a Bunimovich stadium billiard. Our results are twofold. First, we develop a method to solve in a simple way the quantum mechanical evolution of planar billiards with moving walls. The formalism is based on the {\it scaling method} \cite{ver} which enables the resolution of the problem in terms of quantities defined over the boundary of the billiard. The second result is related to the quantum aspects of dissipation in systems with complex spectra. We conclude that in a slowly varying evolution the energy is transferred from the boundary to the particle through Landau$-$Zener transitions.

chao-dyn↗
arXiv1995

Instantons in Burgers Equation

The instanton solution for the forced Burgers equation is found. This solution describes the exponential tail of the probability distribution function of velocity differences in the region where shock waves are absent. The results agree with the one found recently by Polyakov, who used the operator product conjecture. If this conjecture is true, then our WKB asymptotics of the Wyld functional integral is exact to all orders of the perturbative expansion around the instanton solution. We explicitly checked this in the first order. We also generalized our solution for the arbitrary dimension of Burgers (=KPZ) equation. As a result we found the angular dependence of the velocity difference PDF.

hep-th↗
arXiv1996

Structure of Complex-Periodic and Chaotic Media with Spiral Waves

The spatiotemporal structure of reactive media supporting a solitary spiral wave is studied for systems where the local reaction law exhibits a period-doubling cascade to chaos. This structure is considerably more complex than that of simple period-1 oscillatory media. As one moves from the core of the spiral wave the local dynamics takes the form of perturbed period-doubled orbits whose character varies with spatial location relative to the core. An important feature of these media is the existence of a curve where the local dynamics is effectively period-1. This curve arises as a consequence of the necessity to reconcile the conflict between the global topological organization of the medium induced by the presence of a spiral wave and the topological phase space structure of local orbits determined by the reaction rate law. Due to their general topological nature, the phenomena described here should be observable in a broad class of systems with complex periodic behavior.

patt-sol↗
arXiv1998

Temporal intermittency and cascades in shell models of turbulence

The 2d and 3d like Gletzer, Okhitani and Yamada (GOY) shell models are examined. The 2d like model shows a transition from statistical quasi-equilibrium to cascade of enstrophy as a function of the spectral ratio of energy to enstrophy. The transition is related to the ratio of time scales, corresponding to eddy turnover times, between shells. The anomalous scaling, giving rise to non-linear scaling functions, is also connected to the ratio of eddy turnover times. This is illustrated in a simple stochastic model, where the structure function, $ζ(q)$, becomes independent of $q$. In the 3d like model the multiscaling is also influenced by the existence of a second non-positive definite inviscid invariant, the helicity.

chao-dyn↗
arXiv1996

Periodic Orbits and Spectral Statistics of Pseudointegrable Billiards

We demonstrate for a generic pseudointegrable billiard that the number of periodic orbit families with length less than $l$ increases as $πb_0l^2/\langle a(l) \rangle$, where $b_0$ is a constant and $\langle a(l) \rangle$ is the average area occupied by these families. We also find that $\langle a(l) \rangle$ increases with $l$ before saturating. Finally, we show that periodic orbits provide a good estimate of spectral correlations in the corresponding quantum spectrum and thus conclude that diffraction effects are not as significant in such studies.

chao-dyn↗
arXiv1996

Controlling spatiotemporal dynamics with time-delay feedback

We suggest a spatially local feedback mechanism for stabilizing periodic orbits in spatially extended systems. Our method, which is based on a comparison between present and past states of the system, does not require the external generation of an ideal reference state and can suppress both absolute and convective instabilities. As an example, we analyze the complex Ginzburg-Landau equation in one dimension, showing how the time-delay feedback enlarges the stability domain for travelling waves.

chao-dyn↗
arXiv1996

Roundoff-induced Coalescence of Chaotic Trajectories

Numerical experiments recently discussed in the literature show that identical nonlinear chaotic systems linked by a common noise term (or signal) may synchronize after a finite time. We study the process of synchronization as function of precision of calculations. Two generic behaviors of the average coalescence time are identified: exponential or linear. In both cases no synchronization occurs if iterations are done with {\em infinite} precision.

chao-dyn↗
arXiv1999

A Tool to Recover Scalar Time-Delay Systems from Experimental Time Series

We propose a method that is able to analyze chaotic time series, gained from exp erimental data. The method allows to identify scalar time-delay systems. If the dynamics of the system under investigation is governed by a scalar time-delay differential equation of the form $dy(t)/dt = h(y(t),y(t-τ_0))$, the delay time $τ_0$ and the functi on $h$ can be recovered. There are no restrictions to the dimensionality of the chaotic attractor. The method turns out to be insensitive to noise. We successfully apply the method to various time series taken from a computer experiment and two different electronic oscillators.

chao-dyn↗
arXiv1996

Generation and Structure of Solitary Rossby Vortices in Rotating Fluids

The formation of zonal flows and vortices in the generalized Charney-Hasegawa-Mima equation is studied. We focus on the regime when the size of structures is comparable to or larger than the deformation (Rossby) radius. Numerical simulations show the formation of anticyclonic vortices in unstable shear flows and ring-like vortices with quiescent cores and vorticity concentrated in a ring. Physical mechanisms that lead to these phenomena and their relevance to turbulence in planetary atmospheres are discussed.

chao-dyn↗
arXiv1996

Synchronization, chaos, and breakdown of collective domain oscillations in reaction-diffusion systems

The universal equations describing collective oscillations of the multidomain patterns of small period in an arbitrary $d$-dimensional reaction-diffusion system of the activator-inhibitor type are asymptotically derived. It is shown that not far from the instability leading to the formation of the pulsating multidomain pattern the oscillations of different domains synchronize. In one dimension standing and traveling waves of the oscillation phase are realized. In addition to these, in two dimensions target and spiral waves of the oscillation phase, as well as spatio-temporal chaos of domain oscillations, are feasible. Further inside the unstable region the collective oscillations break down, so the pulsating multidomain pattern transforms into an irregular pulsating pattern, the uniform self-oscillations, or turbulence. The parameter regions where these effects occur are analyzed. The effects of the pattern's disorder are also studied. The conclusions of the analysis are supported by the numerical simulations of a concrete model. The obtained results explain the dynamics of Turing patterns observed in the experiments on chlorite-iodide-malonic acid reaction.

patt-sol↗
arXiv1996

Uniform approximation for diffractive contributions to the trace formula in billiard systems

We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional billiard systems with corners. This is achieved by using the exact Sommerfeld solution for the Green function of a wedge. We obtain a uniformly valid formula which interpolates between formerly separate approaches (the geometrical theory of diffraction and Gutzwiller's trace formula). It yields excellent numerical agreement with exact quantum results, also in cases where other methods fail.

chao-dyn↗
arXiv1996

Nonuniversality in level dynamics

Statistical properties of parametric motion in ensembles of Hermitian banded random matrices are studied. We analyze the distribution of level velocities and level curvatures as well as their correlation functions in the crossover regime between three universality classes. It is shown that the statistical properties of level dynamics are in general non-universal and strongly depend on the way in which the parametric dynamics is introduced.

chao-dyn↗
arXiv1996

Viscosity of Colloidal Suspensions

Simple expressions are given for the Newtonian viscosity $η_N(ϕ)$ as well as the viscoelastic behavior of the viscosity $η(ϕ,ω)$ of neutral monodisperse hard sphere colloidal suspensions as a function of volume fraction $ϕ$ and frequency $ω$ over the entire fluid range, i.e., for volume fractions $0 < ϕ< 0.55$. These expressions are based on an approximate theory which considers the viscosity as composed as the sum of two relevant physical processes: $η(ϕ,ω) = η_{\infty}(ϕ) + η_{cd}(ϕ,ω)$, where $η_{\infty}(ϕ) = η_0 χ(ϕ)$ is the infinite frequency (or very short time) viscosity, with $η_0$ the solvent viscosity, $χ(ϕ)$ the equilibrium hard sphere radial distribution function at contact, and $η_{cd}(ϕ,ω)$ the contribution due to the diffusion of the colloidal particles out of cages formed by their neighbors, on the Péclet time scale $τ_P$, the dominant physical process in concentrated colloidal suspensions. The Newtonian viscosity $η_N(ϕ) = η(ϕ,ω= 0)$ agrees very well with the extensive experiments of Van der Werff et al and others. Also, the asymptotic behavior for large $ω$ is of the form $η_{\infty}(ϕ) + A(ϕ)(ωτ_P)^{-1/2}$, in agreement with these experiments, but the theoretical coefficient $A(ϕ)$ differs by a constant factor $2/χ(ϕ)$ from the exact coefficient, computed from the Green-Kubo formula for $η(ϕ,ω)$. This still enables us to predict for practical purposes the visco-elastic behavior of monodisperse spherical colloidal suspensions for all volume fractions by a simple time rescaling.

chao-dyn↗
arXiv1997

Testing the Markov condition in ion channel recordings

A statistical test is presented to decide whether data are adequately described by probabilistic functions of finite state Markov chains (''hidden Markov models'') as applied in the analysis of ion channel data. Particularly, the test can be used to decide whether a system obeys the Markov condition. Simulation studies are performed in order to investigate the sensitivity of the proposed test against violations of the model assumptions. The test can be applied analogously to Markov models.

chao-dyn↗
arXiv1996

Nonhermitean Random Matrix Models : a Free Random Variable Approach

Using the standard concepts of free random variables, we show that for a large class of nonhermitean random matrix models, the support of the eigenvalue distribution follows from their hermitean analogs using a conformal transformation. We also extend the concepts of free random variables to the class of nonhermitean matrices, and apply them to the models discussed by Ginibre-Girko (elliptic ensemble) and Mahaux-Weidenmüller (chaotic resonance scattering).

hep-ph↗
arXiv1996

Studies of Phase Turbulence in the One Dimensional Complex Ginzburg-Landau Equation

The phase-turbulent (PT) regime for the one dimensional complex Ginzburg-Landau equation (CGLE) is carefully studied, in the limit of large systems and long integration times, using an efficient new integration scheme. Particular attention is paid to solutions with a non-zero phase gradient. For fixed control parameters, solutions with conserved average phase gradient $ν$ exist only for $|ν|$ less than some upper limit. The transition from phase to defect-turbulence happens when this limit becomes zero. A Lyapunov analysis shows that the system becomes less and less chaotic for increasing values of the phase gradient. For high values of the phase gradient a family of non-chaotic solutions of the CGLE is found. These solutions consist of spatially periodic or aperiodic waves travelling with constant velocity. They typically have incommensurate velocities for phase and amplitude propagation, showing thereby a novel type of quasiperiodic behavior. The main features of these travelling wave solutions can be explained through a modified Kuramoto-Sivashinsky equation that rules the phase dynamics of the CGLE in the PT phase. The latter explains also the behavior of the maximal Lyapunov exponents of chaotic solutions.

chao-dyn↗