I. A. Khovanov

The effect of noise on strange nonchaotic attractors.

The dynamical response of an underdamped Duffing oscillator to a quasiperiodic force is investigated in the presence and absence of very weak additive noise. Particular attention is focused on the effect of noise on the characteristics of strange nonchaotic attractors (SNAs). It is concluded that even extremely weak noise is sufficient to induce dynamical complexity in an SNA.

Fluctuational escape from a quasi-hyperbolic attractor in the Lorenz system

Noise-induced escape from the basin of attraction of a quasi-hyperbolic chaotic attractor in the Lorenz system is considered. The investigation is carried out in terms of the theory of large fluctuations by experimentally analyzing the escape prehistory. The optimal escape trajectory is shown to be unique and determined by the saddle-point manifolds of the Lorenz system. We established that the escape process consists of three stages and that noise plays a fundamentally different role at each of these stages. The dynamics of fluctuational escape from a quasi-hyperbolic attractor is shown to differ fundamentally from the dynamics of escape from a nonhyperbolic attractor considered previously [1]. We discuss the possibility of analytically describing large noise-induced deviations from a quasi-hyperbolic chaotic attractor and outline the range of outstanding problems in this field.

Stochastic resonance in passive and active electronic circuits

The phenomenon of stochastic resonance in a bistable system modeling overdamped oscillator is studied by numerical simulations and experiments. Experimental data are compared with theoretical results. Stochastic resonance in Chua’s circuit is investigated in detail for different regimes of its own dynamics. The main characteristics of stochastic resonance for different regimes under the adiabatic approximation are compared.

NOISE-INDUCED ESCAPE FROM THE LORENZ ATTRACTOR

Noise-induced escape from a quasi-hyperbolic attractor in the Lorenz system is investigated via an analysis of the distributions of both the escape trajectories and the corresponding realizations of the random force. It is shown that a unique escape path exists, and that it consists of three parts with noise playing a dierent role in each. It is found that the mechanism of the escape from a quasi-hyperbolic attractor diers from that of escape from a non-hyperbolic attractor. The possibility of calculating the escape probability is discussed.

Mutual synchronization and desynchronization of Lorenz systems

The dynamics of two symmetrically coupled Lorenz systems is investigated by means of a numerical experiment. A bifurcation analysis of the synchronization process is presented. The results are compared with numerical experiments. It is shown that changing the coupling can synchronize or desynchronize the subsystems.

Fluctuational escape from chaotic attractors in multistable systems

Recent progress towards an understanding of fluctuational escape from chaotic attractors (CAs) is reviewed and discussed in the contexts of both continuous systems and maps. It is shown that, like the simpler case of escape from a regular attractor, a unique most probable escape path (MPEP) is followed from a CA to the boundary of its basin of attraction. This remains true even where the boundary structure is fractal. The importance of the boundary conditions on the attractor is emphasized. It seems that a generic feature of the escape path is that it passes via certain unstable periodic orbits. The problems still remaining to be solved are identified and considered.

Experimental studies of the non-adiabatic escape problem

Noise-induced transitions between coexisting stable states of a periodically driven nonlinear oscillator have been investigated by means of analog experiments and numerical simulations in the non-adiabatic limit for a wide range of oscillator parameters. It is shown that, for over-damped motion, the field-induced corrections to the activation energy can be described quantitatively in terms of the logarithmic susceptibility (LS) and that the measured frequency dispersion of the corresponding corrections for a weakly damped nonlinear oscillator is in qualitative agreement with the theoretical prediction. Resonantly directed diffusion is observed in numerical simulations of a weakly damped oscillator. The possibility of extending the LS approach to encompass escape from the basin of attraction of a quasi-attractor is discussed.

Mutual synchronization of switching processes in coupled Lorenz systems

A numerical analysis is made of the synchronization of the mean switching frequencies in two symmetrically coupled Lorenz systems functioning in a chaotic regime. The observed effect on the coupling-mismatch parameter plane corresponds to a region of synchronization of the switching processes, within which the mean switching frequencies coincide to a given accuracy.

Fluctuational Escape from Chaotic Attractors

Noise-induced escape from a non-hyperbolic attractor, and from a quasihyperbolic attractor with nonfractal boundaries, is investigated by means of analogue experiments and numerical simulations. It is found that there exists a most probable (optimal) escape trajectory, the prehistory of the escape being defined by the structure of the chaotic attractor. The corresponding optimal fluctuational force is found. The possibility of achieving analytic estimates of the escape probability within the framework of Hamiltonian formalism is discussed.

Nonlinear signal transformation in the regime of SR

We discuss the peculiarities of stochastic resonance in a bistable system for different levels of the amplitude of a regular force. It is shown that for a certain class of signals the output signal-to-noise ratio, as conventionally defined, can substantially exceed the signal-to-noise ratio at the input. The connection between the phenomenon of switching synchronization by regular force and the improvement of the signal-to-noise ratio is considered.