Deterministic Chaos

Spectra of oscillations in the system of two coupled self-sustained chaotic oscillators are investigated in present work. The relation between spectra and partial effective phase diffusion coefficients is determined. We follow the evolution of spectra and diffusion coefficients from the asynchronous regime to the regime of synchronous chaos. The analogy between spectral characteristics of coupled chaotic oscillators and noisy coupled periodic oscillators is drawn.

In present paper the following effects in nonlinear oscillator with final dissipation are studied: stochastic resonance, stochastic synchronization and noise-induced chaos. It is shown that stochastic resonance and stochastic synchronization at final dissipation have the same regularities as in the case of overdamped oscillator but are observed at a lower noise level.

Problems in describing chaotic phase synchronization are connected with ambiguity of definition of istantaneous phase as well as with limiting of its applicability by the coherent chaos regime. We demonstrate that this phenomenon can be analysed by means of function of mutual coherence which has not these restrictions.

In the work the influence of time delay of coupling on the complete synchronization of chaos in an interacting systems with discrete time is studied. The system’s behavior is considered in dependence on coupling coefficient value and delay time value. It is established that coupling with time delay prevents appearance of the complete synchronization of chaos, however it allows the synchronization of periodic and quasi-periodic oscillations.

In this paper we studied intermittent modes in the two-parametric set of onedimensional maps with the neutral unstable point at a phase space boundary. We built the phase diagram in a space of parameters. It defines possible transitions to chaos with a parameter change. We showed the unusual mode of the intermittency concurrence. We studied the laminar length distribution function, Lyapunov exponent and topological entropy of this maps set.

It is shown that the structure of the invariant density for R´ enyi map xn+1 = bxn mod 1, (1 < b < 2) may be clarified by action of the Perron–Frobenius operator on the uniform distribution. The invariant density is presented by finite linear combination of characteristic functions defined on the unit interval according to special rule. Some algebraic equations with entire coefficients are formulated for parameter b corresponding values definition.

The influence of two types of diffusion on dynamics of stochastic lattice Lotka–Volterra model is considered in this work. The simulations were carried out by means of Kinetic Monte-Carlo algorithm. It is shown that the local diffusion considerably changes 75the dynamics of the model and accelerates the interaction processes on the lattice, while the mixing results in appearance of global periodic oscillations. The global oscillations appear due to phenomenon of phase synchronization.

We consider the Arnol’d diffusion in a system with 2.5 degrees of freedom along a resonance with an external oscillating field. The analytical estimation of the diffusion coefficient we made is in a good agreement with numerical results. It’s also shown that both the amplitude of external field and the parameter of weak interaction between two spatial degrees of freedom have an influence on Arnol’d diffusion manifestation and its rate.

Chaotic dynamics of a system of four nonlinear coupled non-identical Landau-Stuart oscillators is considered. Subsystems are activated alternately by pairs due to aslow variation of their parameters responsible for the Andronov–Hopf bifurcation. It is shown, that system dynamics depends of coupling type. Different types of phase map (Bernoulli type map) are obtained in Poincar´ e section depending of coupling. Some systems with different type of coupling corresponded to «maximum» and «minimum» chaos are investigated.

The method for controlling chaos in a ring resonator filled with a medium with cubic phase nonlinearity (Ikeda system), suggested in [1], is investigated within the framework of a distributed spatio-temporal model described by a Nonlinear Schr¨ odinger Equation with time-delayed boundary condition. Numerical results are presented which confirm the capability of the suggested method. For the case of weakly dispersive nonlinear medium, the results are in good agreement with the approximate theory based on the return map [1].