Instability and Dynamical Chaos in a Weak Nonlinear Interaction of Waves * (original) (raw)

Stochastic Resonance in Chaotic Dynamics

It is suggested that chaotic dynamical systems characterized by intermittent jumps between two preferred regions of phase space display an enhanced sensitivity to weak periodic forcings through a stochastic resonance-like mechanism. This possibility is illustrated by the study of the residence time distribution in two examples of bi-modal chaos: the periodically forced Duffing oscillator and a 1-dimensional map showing intermittent behavior.

Chaotic saddles at the onset of intermittent spatiotemporal chaos

Physical Review E, 2007

In a recent study ͓Rempel and Chian, Phys. Rev. Lett. 98, 014101 ͑2007͔͒, it has been shown that nonattracting chaotic sets ͑chaotic saddles͒ are responsible for intermittency in the regularized long-wave equation that undergoes a transition to spatiotemporal chaos ͑STC͒ via quasiperiodicity and temporal chaos. In the present paper, it is demonstrated that a similar mechanism is present in the damped Kuramoto-Sivashinsky equation. Prior to the onset of STC, a spatiotemporally chaotic saddle coexists with a spatially regular attractor. After the transition to STC, the chaotic saddle merges with the attractor, generating intermittent bursts of STC that dominate the post-transition dynamics.

Spatiotemporal intermittency and chaotic saddles in the regularized long-wave equation

Physics of Fluids, 2009

Transition to intermittent spatiotemporal chaos is studied in the regularized long-wave equation, a nonlinear model of shallow water waves. A mechanism for the onset of on-off spatiotemporal intermittency is explored. In this mechanism, the coupling of two chaotic saddles triggers random switching between phases of laminar and bursty behaviors. The average time between bursts as a function of the control parameter follows a power law typical of crisis transitions in chaotic systems. The degree of spatiotemporal disorder in the observed fluid patterns is quantified by means of the time-averaged spectral entropy for both chaotic attractors and chaotic saddles. The implications of these results to other fluid systems are discussed.

The evolution of spatiotemporal chaos in a discrete-continuous active medium

Technical Physics Letters, 2017

A special approach to calculation of the spectrum of Lyapunov exponents has been developed and applied to diagnostics of the degree of regularity of wave structures in a model neural system. A transition between the regimes of regular wave dynamics and developed spatiotemporal chaos is demonstrated and quantitatively characterized.

Stochastic resonance in deterministic chaotic systems The mechanism of stochastic resonance

A Crisanti, M Falcioni, G Lacorata et al. Predictability of velocity and temperature fields in intermittent turbulence A Crisanti, M H Jensen, G Paladin et al. Stochastic resonance: noise-enhanced order Vadim S Anishchenko, Arkady B Neiman, F Moss et al. Predictability in spatially extended systems G Paladin and A Vulpiani Regular and chaotic motion of fluid particles in a two-dimensional fluid M Falcioni, G Paladin and A Vulpiani Analogue studies of nonlinear systems D G Luchinsky, P V E McClintock and M I Dykman Fractal measures in the random-field Ising model S N Evangelou The mechanism of stochastic resonance R Benzi, A Sutera and A Vulpiani Characterisation of intermittency in chaotic systems R Benzi, G Paladin, G Parisi et al. Abstract. We propose a mechanism which produces periodic variarions of the degree of prediaability in dynamical systems. It is shown that even in the absence of noise when the control parameter changes periodically in time, below and above the m o l d for the onset of chaos, stochastic resonance effects appear. As a result one has an alternation of chaotic and regular. i.e. predictable, evolutions in m dmst periodic way, so that the Lyapnnov exponent is positive but some time correlations do not d a y .

Effect of Chaos and Stochastics Induced By Internal Waves on Acoustic Wave Propagation

Office of the Provost …, 2005

In the present paper the eikonal equation is considered in the form of a second order, nonlinear ordinary differential equation with harmonic excitation due to internal wave. The harmonic excitation is taken imperfect, i.e. there is a random phase modulation due to Gaussian white noise. The amplitude and wavelength of the acoustic wave are used as the principle signal characteristics in bifurcation analysis. The regions of instability, identified using the bifurcation diagrams, examined through phase diagrams and Poincare maps. The effect of stochastic nature in addition to chaotic one of internal waves is demonstrated by means of comparison of the numerical data obtained for perfectly periodic excitation. Preliminary analysis shows very strong dependence on noise intensity at some values of amplitude and wave length of the internal wave.

Correlation analysis of dynamical chaos

2003

We study correlation and spectral properties of chaotic self-sustained oscillations of di erent types. It is shown that some classical models of stochastic processes can be used to describe behavior of autocorrelation functions of chaos. The in uence of noise on chaotic systems is also considered.

Synchronization of Chaos and the Transition to Wave Turbulence

International Journal of Bifurcation and Chaos, 2012

We investigated the transition to wave turbulence in a spatially extended three-wave interacting model, where a spatially homogeneous state undergoing chaotic dynamics undergoes spatial mode excitation. The transition to this weakly turbulent state can be regarded as the loss of synchronization of chaos of mode oscillators describing the spatial dynamics.

Chaotic motion and stochastic excitation

Nonlinear Dynamics, 1994

In this paper, one considers dynamic chaotic/stochastic systems and the associated Gibbs set. The behavior of these sets leads one to characterize the systems and to calculate the values of the Kolmogorov entropy. The ultimate objective is to extend an approach typical of the statistical mechanics to the analysis of systems of the mechanical engineering.

CHAOTIC DYNAMICS OF A SPATIO-INHOMOGENEOUS MEDIUM

International Journal of Bifurcation and Chaos, 2005

In the present paper we show that inhomogeneity of a continuous self-sustained oscillating medium can be a reason for the onset of chaotic behavior. It has been established that temporal chaotic dynamics typically arises in the medium with a linear mismatch of the natural frequency along a spatial coordinate, whereas a chaotic regime is not characteristic for the medium with randomly distributed frequencies. The interconnection has been revealed between the temporal chaotic behavior and the spatial formation of imperfect clusters. The spectral and correlation analysis as well as the linear analysis of stability of regular and chaotic regimes in the inhomogeneous medium are performed. The correlation of the instantaneous phase dynamics of oscillations with the behavior of autocorrelation functions has been examined. It has been established that the characteristics of temporal chaos correspond to a spiral attractor (Shilnikov's attractor).