Tatjana E. Vadivasova

Chaotic attractors of two-dimensional invertible maps

In this paper, we investigate the characteristics of quasihyperbolic attractors and quasiattractors in Invertible dissipative maps of the plane. The criteria which allow one to diagnose the indicated types of attractors in numerical experiments are formulated.

INSTANTANEOUS PHASE METHOD IN STUDYING CHAOTIC AND STOCHASTIC OSCILLATIONS AND ITS LIMITATIONS

We study the behavior of an instantaneous phase and mean frequency of chaotic self-sustained oscillations and noise-induced stochastic oscillations. The results obtained by using various methods of the phase definition are compared to each other. We also compare two methods for describing synchronization of chaotic self-sustained oscillations, namely, instantaneous phase locking and locking of characteristic frequencies in power spectra. It is shown that the technique for diagnostics of the chaos synchronization based on the instantaneous phase locking is not universal.

SPECTRAL AND CORRELATION ANALYSIS OF SPIRAL CHAOS

We study numerically the behavior of the autocorrelation function (ACF) and the power spectrum of spiral attractors without and in the presence of noise. It is shown that the ACF decays exponentially and has two different time scales. The rate of the ACF decrease is defined by the amplitude fluctuations on small time intervals, i.e., when τ < τcor, and by the effective diffusion coefficient of the instantantaneous phase on large time intervals. It is also demonstrated that the ACF in the Poincare map also decreases according to the exponential law exp(- λ+ k), where λ+ is the positive Lyapunov exponent. The obtained results are compared with the theory of fluctuations for the Van der Pol oscillator.

Experimental Studies of Noise Effects in Nonlinear Oscillators

In the paper the noisy behavior of nonlinear oscillators is explored experimentally. Two types of excitable stochastic oscillators are considered and compared, i.e., the FitzHugh–Nagumo system and the Van der Pol oscillator with a subcritical Andronov–Hopf bifurcation. In the presence of noise and at certain parameter values both systems can demonstrate the same type of stochastic behavior with effects of coherence resonance and stochastic synchronization. Thus, the excitable oscillators of both types can be classified as stochastic self-sustained oscillators. Besides, the noise influence on a supercritical Andronov–Hopf bifurcation is studied. Experimentally measured joint probability distributions enable to analyze the phenomenological stochastic bifurcations corresponding to the boundary of the noisy limit cycle regime. The experimental results are supported by numerical simulations.

About stationary probability measure of nonhyperbolic attractors

In the present paper we make an attempt to give evidence of the existence of stationary probability measure of nonhyperbolic attractors in the presence of noise. We analyze 2-dimensional invertible maps by using methods of stochastic equations and of evolution equations for probability density. We show that these approaches are adequate also for nonlinear systems with nonhyperbolic attractors.

Peculiarities of Nonhyperbolic Chaos

In this paper we study properties of hyperbolic and nonhyperbolic attractors. On the basis of the method proposed in [1] we present a numerical procedure to distinguish two types of chaotic attractors in two-dimensional (2-dim) invertible maps and in three-dimensional (3-dim) flow systems. We also analyze the effect of bounded noise on certain characteristics of nonhyperbolic chaos. We compute the stationary probability measure on noisy nonhyperbolic attractors by means of two different methods and then compare the obtained results.