Influence of noise on statistical properties of nonhyperbolic attractors (original) (raw)
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Stability Properties of Nonhyperbolic Chaotic Attractors under Noise
2004
We study local and global stability of nonhyperbolic chaotic attractors contaminated by noise. The former is given by the maximum distance of a noisy trajectory from the noisefree attractor, while the latter is provided by the minimal escape energy necessary to leave the basin of attraction, calculated with the Hamiltonian theory of large fluctuations. We establish the important and counterintuitive result that both concepts may be opposed to each other. Even when one attractor is globally more stable than another one, it can be locally less stable. Our results are exemplified with the Holmes map, for two different sets of parameter, and with a juxtaposition of the Holmes and the Ikeda maps. Finally, the experimental relevance of these findings is pointed out.
Stability Properties of Nonhyperbolic Chaotic Attractors with Respect to Noise
Physical Review Letters, 2004
We study local and global stability of nonhyperbolic chaotic attractors contaminated by noise. The former is given by the maximum distance of a noisy trajectory from the noisefree attractor, while the latter is provided by the minimal escape energy necessary to leave the basin of attraction, calculated with the Hamiltonian theory of large fluctuations. We establish the important and counterintuitive result that both concepts may be opposed to each other. Even when one attractor is globally more stable than another one, it can be locally less stable. Our results are exemplified with the Holmes map, for two different sets of parameter, and with a juxtaposition of the Holmes and the Ikeda maps. Finally, the experimental relevance of these findings is pointed out.
Physical Review Letters, 2001
We study the influence of external noise on the relaxation to an invariant probability measure for two types of nonhyperbolic chaotic attractors, a spiral (or coherent) and a noncoherent one. We find that for the coherent attractor the rate of mixing changes under the influence of noise, although the largest Lyapunov exponent remains almost unchanged. A mechanism of the noise influence on mixing is presented which is associated with the dynamics of the instantaneous phase of chaotic trajectories. This also explains why the noncoherent regime is robust against the presence of external noise.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2002
We study the relaxation to an invariant probability measure on quasihyperbolic and nonhyperbolic chaotic attractors in the presence of noise. We also compare different characteristics of the rate of mixing and show numerically that the rate of mixing for nonhyperbolic chaotic attractors can significantly change under the influence of noise. A mechanism of the noise influence on mixing is presented, which is associated with the dynamics of the instantaneous phase of chaotic trajectories. We also analyze how the synchronization effect can influence the rate of mixing in a system of two coupled chaotic oscillators.
Effects of noise on a nonuniform chaotic map
Physics Letters A, 1987
Nonuniform separation of orbits initially close to each other is measured by several quantities which are derived from the statistics ofgrowth rates ofsmall perturbations. Using these measuresof nonuniformity, a Belousov-Zhabotinsky map (BZ map), the logistic map, and the tent map are compared. The extremely nonuniform BZ map shows a remarkable response to external noise: the state predictability can be improved by an increase in noise power.
Statistical properties of dynamical chaos
Mathematical Biosciences and Engineering, 2004
This study presents a survey of the results obtained by the authors on statistical description of dynamical chaos and the effect of noise on dynamical regimes. We deal with nearly hyperbolic and nonhyperbolic chaotic attractors and discuss methods of diagnosing the type of an attractor. We consider regularities of the relaxation to an invariant probability measure for different types of attractors. We explore peculiarities of autocorrelation decay and of power spectrum shape and their interconnection with Lyapunov exponents, instantaneous phase diffusion and the intensity of external noise. Numeric results are compared with experimental data.
About asymmetric noisy chaotic maps
In this paper we will introduce three families of asymmetric maps, and discuss some dynamical properties for these families in the deterministic case, and noisy case. New mixed noisy chaotic map will be suggested and then studied with some details.
Noise-induced unstable dimension variability and transition to chaos in random dynamical systems
Physical Review E - PHYS REV E, 2003
Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapu...
Statistical approach to nonhyperbolic chaotic systems
Spectral properties of the evolution operator for probability densities are obtained for unimodal maps for which all periodic orbits are unstable, and the Lyapunov exponent calculated from the first iterate of the critical point converges to a positive constant. The method is applied to the logistic map both for parameter values at which finite Markov partitions can be found as well as for more typical parameter values. A universal behavior is found for the spectral gap in the period-doubling inverse cascade of chaotic band-merging bifurcations. Full agreement with numerical simulation is obtained.
The influence of noise on the correlation dimension of chaotic attractors
1998
The present paper investigates the influence of noise on the correlation dimension Dc of chaotic attractors arising in discrete and continuous in time dynamical systems. Our numerical results indicate that the presence of noise leads to an increase of the correlation dimension. Assuming that the correlation dimension for a white noise is infinite, we prove, first, that the increase of the dimension of a chaotic attractor in a stochastic system is a generic property of the set of stochastic dynamical systems and, secondly, that the existence of a small correlation dimension in a time series implies that the deterministic part of its Wold decomposition is nonzero. We also present a collection of dynamical systems subject to noise which may be considered as models for predictions on the response of time series with a finite correlation dimension, as encountered in physical or numerical experiments.