Vadim S. Anishchenko

Nonlinear dynamics of chaotic and stochastic systems : tutorial and modern developments

From the contents: Tutorial * Dynamical Systems * Fluctuations in Dynamic Systems * Synchronization of Periodic Systems * Dynamical Chaos * Routes to Chaos * Synchronization of Chaos * Controlling Chaos * Reconstruction of Dynamical Systems * Stochastic Dynamics * Stochastic Resonance * Synchronization of Stochastic Systems * The Beneficial Role of Noise in Excitable Systems * Noise Induced Transport.

SYNCHRONIZATION OF CHAOS

This paper is devoted to the problem of synchronization of dynamical systems in chaotic oscillations regimes. The authors attempt to use the ideas of synchronization and its mechanisms on a certain class of chaotic oscillations. These are chaotic oscillations for which one can pick out basic frequencies in their power spectra. The physical and computer experiments were carried out for the system of two coupled auto-oscillators. The experimental installation permitted one to realize both unidirectional coupling (external synchronization) and symmetrical coupling (mutual synchronization). An auto-oscillator with an inertial nonlinearity was chosen as a partial subsystem. It possesses a chaotic attractor of spiral type in its phase space. It is known that such chaotic oscillations have a distinguished peak in the power spectrum at the frequency f0 (basic frequency). In the experiments, one could make the basic frequencies of partial oscillators equal or different. The bifurcation diagrams on the plane of con...

Stochastic resonance in chaotic systems

The phenomenon of stochastic resonance (SR) is investigated for chaotic systems perturbed by white noise and a harmonic force. The bistable discrete map and the Lorenz system are considered as models. It is shown that SR in chaotic systems can be realized via both parameter variation (in the absence of noise) and by variation of the noise intensity with fixed values of the other parameters.

STOCHASTIC RESONANCE IN CHUA’S CIRCUIT

In this paper, we report numerical observations of the stochastic resonance (SR) phenomenon in a bistable chaotic electronic circuit (namely, Chua’s circuit) driven simultaneously by noise and a sinusoidal signal. It is shown that the noise-induced “chaos-chaos” type intermittency is a physical mechanism of the SR-phenomenon in chaotic systems. The resulting amplification of the sinusoidal signal intensity is due to a coherent interaction of three characteristic frequencies of the system. The SR-phenomenon can be controlled by a variation of either the noise intensity or the system parameters in the absence of noise.

STOCHASTIC RESONANCE IN CHUA’S CIRCUIT DRIVEN BY AMPLITUDE OR FREQUENCY MODULATED SIGNALS

Using numerical simulation, we establish the possibility of realizing the stochastic resonance (SR) phenomenon in Chua’s circuit when it is excited by either an amplitude-modulated or a frequency-modulated signal. It is shown that the application of a frequency-modulated signal to a Chua’s circuit operating in a regime of dynamical intermittency is preferable over an amplitude-modulated signal from the point of view of minimizing the signal distortion and maximizing the signal-to-noise ratio (SNR).

Coherence-Resonance Chimeras in a Network of Excitable Elements.

We demonstrate that chimera behavior can be observed in nonlocally coupled networks of excitable systems in the presence of noise. This phenomenon is distinct from classical chimeras, which occur in deterministic oscillatory systems, and it combines temporal features of coherence resonance, i.e., the constructive role of noise, and spatial properties of chimera states, i.e., the coexistence of spatially coherent and incoherent domains in a network of identical elements. Coherence-resonance chimeras are associated with alternating switching of the location of coherent and incoherent domains, which might be relevant in neuronal networks.

Phase dynamics of two coupled oscillators under external periodic force

The effect of synchronization has been studied in a system of two coupled Van der Pol oscillators under external harmonic force. The analysis has been carried out using the phase approach. The mechanisms of complete and partial synchronization have been established. The main type of bifurcation described in this paper is the saddle-node bifurcation of invariant curves that corresponds to the saddle-node bifurcation of two-dimensional tori in the complete system of differential equations for the dynamical system under study.

Synchronization of chaotic oscillators by periodic parametric perturbations

Abstract We show that the synchronization of coupled chaotic oscillators can be achieved by means of periodic parametric perturbations of the coupling element. The possibility of synchronization is demonstrated for the simple case of two identical nonautonomous oscillators with piecewise linear characteristics both analytically and numerically. Using linear analysis we have determined the stability conditions for symmetric oscillations.

Birth of double-double scroll attractor in coupled Chua circuits

We discuss the creation of hyperchaotic attractors in a system of two coupled Chua circuits. Both mutual and unidirectional couplings are considered. Results from chaos synchronization theory allow us to determine chaos-hyperchaos intermittency.

ENTRAINMENT BETWEEN HEART RATE AND WEAK NONINVASIVE FORCING

We demonstrate that the heart rate of a healthy human can be synchronized by means of weak external noninvasive forcing in the form of a sequence of sound and light pulses, being either periodic or aperiodic, the latter forcing given by interbeat intervals of the heart of another subject. The phenomenon of phase locking of n:m type is observed for both situations in about 90% of our experiments. The plot for the ratio of forcing frequency to the average frequency of response versus detuning possesses a plateau and is in agreement with classical synchronization theory.