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arXiv1994

Remarks on the mean field dynamics of networks of chaotic elements

Fluctuations of the mean field of a globally coupled dynamical systems are discussed. The origin of hidden coherence is related with the instability of the fixed point solution of the self-consistent Perron-Frobenius equation. Collective dynamics in globally coupled tent maps are re-examined, both with the help of direct simulation and the Perron-Frobenius equation. Collective chaos in a single band state, and bifurcation against initial conditions in a two-band state are clarified with the return maps of the mean-field, Lyapunov spectra, and also the newly introduced Lyapunov exponent for the Perron-Frobenius equation. Future problems on the collective dynamics are discussed.

chao-dyn↗

arXiv1995

Time Delay Correlations in Chaotic Scattering: Random Matrix Approach

We study the correlations of time delays in a model of chaotic resonance scattering based on the random matrix approach. Analytical formulae which are valid for arbitrary number of open channels and arbitrary coupling strength between resonances and channels are obtained by the supersymmetry method. We demonstrate that the time delay correlation function, though being not a Lorentzian, is characterized, similar to that of the scattering matrix, by the gap between the cloud of complex poles of the $S$-matrix and the real energy axis.

chao-dyn↗

arXiv1995

How the viscous subrange determines inertial range properties in turbulence shell models

We calculate static solutions of the 'GOY' shell model of turbulence and do a linear stability analysis. The asymptotic limit of large Reynolds numbers is analyzed. A phase diagram is presented which shows the range of stability of the static solution. We see an unexpected oscillatory dependence of the stability range upon $\lg ν$, where $ν$ is the viscosity. This effect depends upon the discrete structure of the shell model and goes to zero as the separation between the shells is brought to zero. These findings show how viscous effects play a role in determining inertial properties of shell models and give some hints for understanding the effects of viscous dissipation upon real turbulence.

chao-dyn↗

arXiv1996

Study of chaos in hamiltonian systems via convergent normal forms

We use Moser's normal forms to study chaotic motion in two-degree hamiltonian systems near a saddle point. Besides being convergent, they provide a suitable description of the cylindrical topology of the chaotic flow in that vicinity. Both aspects combined allowed a precise computation of the homoclinic interaction of stable and unstable manifolds in the full phase space, rather than just the Poincaré section. The formalism was applied to the Hénon-Heiles hamiltonian, producing strong evidence that the region of convergence of these normal forms extends over that originally established by Moser.

chao-dyn↗

arXiv1995

Quantum Surface of Section Method: Eigenstates and Unitary Quantum Poincare Evolution

The unitary representation of exact quantum Poincare mapping is constructed. It is equivalent to the compact representation in a sense that it yields equivalent quantization condition with important advantage over the compact version: since it preserves the probability it can be literally interpreted as the quantum Poincare mapping which generates quantum time evolution at fixed energy between two successive crossings with surface of section (SOS). SOS coherent state representation (SOS Husimi distribution) of arbitrary (either stationary or evolving) quantum SOS state (vector from the Hilbert space over the configurational SOS) is introduced. Dynamical properties of SOS states can be quantitatively studied in terms of the so called localization areas which are defined through information entropies of their SOS coherent state representations. In the second part of the paper I report on results of extensive numerical application of quantum SOS method in a generic but simple 2-dim Hamiltonian system, namely semiseparable oscillator. I have calculated the stretch of 13500 consecutive eigenstates with the largest sequential quantum number around 18 million and obtained the following results: (i) the validity of the semiclassical Berry-Robnik formula for level spacing statistics was confirmed and using the concept of localization area the states were quantitatively classified as regular or chaotic, (ii) the classical and quantum Poincare evolution were performed and compared, and expected agreement was found, (iii) I studied few examples of wavefunctions and particularly, SOS coherent state representation of regular and chaotic eigenstates and analyzed statistical properties of their zeros which were shown on the chaotic component of 2-dim SOS to be uniformly distributed with the cubic repulsion between nearest neighbours.

chao-dyn↗

arXiv1995

Spatiotemporal bifurcations in plasma drift-waves

Experimental data from an experiment on drift--waves in plasma is presented. The experiment provides a space--time diagnostic and has a control parameter that permits the study of the transition from a stable plasma to a turbulent plasma. The biorthogonal decomposition is used to analyse the data. We introduce the notion of complex modulation for two--dimensional systems. We decompose the real physical system into complex modulated monochromatic travelling waves and give a simple model describing the speed doubling observed in the data as the control parameter increases.

chao-dyn↗

arXiv1995

Trace formulae for three-dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard

This paper is devoted to the quantum chaology of three-dimensional systems. A trace formula is derived for compact polyhedral billiards which tessellate the three-dimensional hyperbolic space of constant negative curvature. The exact trace formula is compared with Gutzwiller's semiclassical periodic-orbit theory in three dimensions, and applied to a tetrahedral billiard being strongly chaotic. Geometric properties as well as the conjugacy classes of the defining group are discussed. The length spectrum and the quantal level spectrum are numerically computed allowing the evaluation of the trace formula as is demonstrated in the case of the spectral staircase N(E), which in turn is successfully applied in a quantization condition.

chao-dyn↗

arXiv1994

Numerical Study of Granular Turbulence and the appearance of $k^{-5/3}$ energy spectrum without flow

The vibrated bed of powder, a vessel that mono disperse glass beads fill and a loud speaker shakes, is investigated numerically with the distinct element method, a kind of molecular dynamics. When the bed is heavily shaken, the displacement vectors of powder have the power spectrum with the dependence upon the wave number $k$ as $k^{-5/3}$. The origin of this spectrum is suggested to be the balance between the injected and dissipative energy, analogous to the proposal by Kolmogorov to explain the $k^{-5/3}$ energy spectrum observed in the fluid turbulence. Furthermore, the same spectrum still appears even without flows of powder. Thus Kolmogorov's argument is more universal than believed before.

chao-dyn↗

arXiv1994

Transition to Chaos in a Shell Model of Turbulence

We study a shell model for the energy cascade in three dimensional turbulence at varying the coefficients of the non-linear terms in such a way that the fundamental symmetries of Navier-Stokes are conserved. When a control parameter $ε$ related to the strength of backward energy transfer is enough small, the dynamical system has a stable fixed point corresponding to the Kolmogorov scaling. This point becomes unstable at $ε=0.3843...$ where a stable limit cycle appears via a Hopf bifurcation. By using the bi-orthogonal decomposition, the transition to chaos is shown to follow the Ruelle-Takens scenario. For $ε> 0.3953..$ the dynamical evolution is intermittent with a positive Lyapunov exponent. In this regime, there exists a strange attractor which remains close to the Kolmogorov (now unstable) fixed point, and a local scaling invariance which can be described via a intermittent one-dimensional map.

chao-dyn↗

arXiv1995

Numerical Study of a Lyapunov Functional for the Complex Ginzburg-Landau Equation

We numerically study in the one-dimensional case the validity of the functional calculated by Graham and coworkers as a Lyapunov potential for the Complex Ginzburg-Landau equation. In non-chaotic regions of parameter space the functional decreases monotonically in time towards the plane wave attractors, as expected for a Lyapunov functional, provided that no phase singularities are encountered. In the phase turbulence region the potential relaxes towards a value characteristic of the phase turbulent attractor, and the dynamics there approximately preserves a constant value. There are however very small but systematic deviations from the theoretical predictions, that increase when going deeper in the phase turbulence region. In more disordered chaotic regimes characterized by the presence of phase singularities the functional is ill-defined and then not a correct Lyapunov potential.

cond-mat↗

arXiv1995

Generalized scaling in fully developed turbulence

In this paper we report numerical and experimental results on the scaling properties of the velocity turbulent fields in several flows. The limits of a new form of scaling, named Extended Self Similarity(ESS), are discussed. We show that, when a mean shear is absent, the self scaling exponents are universal and they do not depend on the specific flow (3D homogeneous turbulence, thermal convection , MHD). In contrast, ESS is not observed when a strong shear is present. We propose a generalized version of self scaling which extends down to the smallest resolvable scales even in cases where ESS is not present. This new scaling is checked in several laboratory and numerical experiment. A possible theoretical interpretation is also proposed. A synthetic turbulent signal having most of the properties of a real one has been generated.

chao-dyn↗

arXiv1996

Spatial chaos in weakly dispersive and viscous media: a nonperturbative theory of the driven KdV-Burgers equation

The asymptotic travelling wave solution of the KdV-Burgers equation driven by the long scale periodic driver is constructed. The solution represents a shock-train in which the quasi-periodic sequence of dispersive shocks or soliton chains is interspersed by smoothly varying regions. It is shown that the periodic solution which has the spatial driver period undergoes period doublings as the governing parameter changes. Two types of chaotic behavior are considered. The first type is a weak chaos, where only a small chaotic deviation from the periodic solution occurs. The second type corresponds to the developed chaos where the solution ``ignores'' the driver period and represents a random sequence of uncorrelated shocks. In the case of weak chaos the shock coordinate being repeatedly mapped over the driver period moves on a chaotic attractor, while in the case of developed chaos it moves on a repellor. Both solutions depend on a parameter indicating the reference shock position in the shock-train. The structure of a one dimensional set to which this parameter belongs is investigated. This set contains measure one intervals around the fixed points in the case of periodic or weakly chaotic solutions and it becomes a fractal in the case of strong chaos. The capacity dimension of this set is calculated.

chao-dyn↗

arXiv1996

The Spatio-Temporal Structure of Spiral-Defect Chaos

We present a study of the recently discovered spatially-extended chaotic state known as spiral-defect chaos, which occurs in low-Prandtl-number, large-aspect-ratio Rayleigh-Benard convection. We employ the modulus squared of the space-time Fourier transform of time series of two-dimensional shadowgraph images to construct the structure factor ${S}({\vec k},ω)$. This analysis is used to characterize the average spatial and temporal scales of the chaotic state. We find that the correlation length and time can be described by power-law dependences on the reduced Rayleigh number $ε$. These power laws have as yet no theoretical explanation.

chao-dyn↗

arXiv1996

The Effect of Small-Scale Forcing on Large-Scale Structures in Two-Dimensional Flows

The effect of small scale forcing on large scale structures in $β$-plane two-dimensional (2D) turbulence is studied using long-term direct numerical simulations (DNS). We find that nonlinear effects remain strong at all times and for all scales and establish an inverse energy cascade that extends to the largest scales available in the system. The large scale flow develops strong spectral anisotropy: $k^{-5/3}$ Kolmogorov scaling holds for almost all $ϕ$, $ϕ=\arctan (k_y/k_x)$, except in the small vicinity of $k_x=0$, where Rhines's $k^{-5}$ scaling prevails. Due to the $k^{-5}$ scaling, the spectral evolution of $β$-plane turbulence becomes extremely slow which, perhaps, explains why this scaling law has never before been observed in DNS. Simulations with different values of $β$ indicate that the $β$-effect diminishes at small scales where the flow is nearly isotropic. Thus, for simulations of $\b$-plane turbulence forced at small scales sufficiently removed from the scales where $\b$-effect is strong, large eddy simulation (LES) can be used. A subgrid scale (SGS) parameterization for such LES must account for the small scale forcing that is not explicitly resolved and correctly accommodate two inviscid conservation laws, viz. energy and enstrophy. This requirement gives rise to a new anisotropic stabilized negative viscosity (SNV) SGS representation which is discussed in the context of LES of isotropic 2D turbulence.

chao-dyn↗

arXiv1996

From Random Matrix Theory to Statistical Mechanics - Anyon Gas

Motivated by numerical experiments and studies of quantum systems which are classically chaotic, we take a Random Matrix description of a Hard-sphere gas to Statistical Mechanical description. We apply this to Anyon gas and obtain a formal expression for the momentum distribution. Various limiting situations are discussed and are found in agreement with the well-known results on Hard-sphere gas in low-density regime.

cond-mat↗

arXiv1993

Self-Preservation of Large-Scale Structures in Burgers' Turbulence

We investigate the stability of large-scale structures in Burgers' equation under the perturbation of high wave-number noise in the initial conditions. Analytical estimates are obtained for random initial data with spatial spectral density k^n, n < 1. Numerical investigations are performed for the case n=0, using a parallel implementation of the Fast Legendre Transform.

chao-dyn↗

arXiv1993

Linear Response Theory in Stochastic Resonance

The susceptibility of an overdamped Markov system fluctuating in a bistable potential of general form is obtained by analytic solution of the Fokker-Planck equation (FPE) for low noise intensities. The results are discussed in the context of the LRT theory of stochastic resonance. They go over into recent results (Gang Hu et al {\em Phys. Lett. A} {\bf 172}, 21, 1992) obtained from the FPE for the case of a symmetrical potential, and they coincide with the LRT results (Dykman et al, {\em Phys. Rev. Lett.} {\bf 65}, 2606, 1990; {\em JETP Lett} {\bf 52}, 144, 1990; {\em Phys. Rev. Lett.} {\bf 68}, 2985, 1992) obtained for the general case of bistable systems.

chao-dyn↗

arXiv1998

Dynamical Instabilities and Deterministic Chaos in Ballistic Electron Motion in Semiconductor Superlattices

We consider the motion of ballistic electrons within a superlattice miniband under the influence of an alternating electric field. We show that the interaction of electrons with the self-consistent electromagnetic field generated by the electron current may lead to the transition from regular to chaotic dynamics. We estimate the conditions for the experimental observation of this deterministic chaos and discuss the similarities of the superlattice system with the other condensed matter and quantum optical systems.

cond-mat.mes-hall↗