Applied Problems of Nonlinear Oscillation and Wave Theory

The stabilization of chaos in the Rossler system by external signal is investigated. Different types of external action are considered: both of pulsed and harmonic signal. There are illustrations: charts of dynamical regimes, phase porters, stroboscopic section of Poincare, spectrum of Lyapunov exponents. Comparative analysis of efficiency of stabilization of band chaos and spiral chaos by different signal is carried out. The dependence of synchronization picture on direction of acting pulses is shown.

Self­organized structures after ion­beam irradiation in solid materials have been studied using the method of fractal dimension. General computer method of the scale invariance evaluation for exposed dispersive structures is described. It was demonstrated that structures after irradiation can be characterized by the compatibility of scale invariance properties at different scale levels.

A chaos­based communication scheme allowing simultaneous bidirectional message transmission (Opt. Lett. 32, 403, 2007) is investigated numerically. Incoherent feedback and coupling case is analyzed, which is expected in real long­distance optical communication systems. It is shown that identical synchronization of chaotic laser waveforms and bidirectional message transmission are possible as in the coherent coupling case. However, the chaotic regime at incoherent feedback and coupling is quite different.

This paper deals with the influence of the passive elements on the synchronization in the ensembles of coupled non­identical Bonhoeffer–van der Pol oscillators. With a help of numerical experiment it was demonstrated that the introduction of passive elements may lead to both increase and decrease of global synchronization threshold in the system. These results were confirmed analytically using piecewise linear approximation of the Bonhoeffer–van der Pol model.

We study the process of competition in the two­component model of the immune T­cells ensemble that underpins the selection mechanism of the most efficient T­cell species (clonotypes). We demonstrate the absence of periodic oscillations, determine the regions of coexistence, partial and mutual extinction of clonotypes. Applicability of the mean field approximation is analyzed. The biological implications of the results are discussed.

The particular properties of dynamics are discussed for dissipatively coupled van der Pol oscillators, non­identical in values of parameters controlling the Andronov–Hopf bifurcation and nonlinear dissipation. Possibility of a special synchronization regime in an infinitively long band between oscillator death and quasiperiodic areas is shown for such system. Non­identity of parameters of nonlinear dissipation results in specific form of the boundary of the main synchronization tongue, which looks like the mirror letter S.

A simple autonomous three­dimensional system is introduced that demonstrates quasiperiodic self­oscillations and has as attractor a two­dimensional torus. The computing illustrations of quasiperiodic dynamics are presented: phase portraits, Fourie spectrums, graphics of Lyapunov exponents. The existing of Arnold tongues on the parametric plane and transition from quasiperiodic dynamics to chaos through destruction of invariant curve in the Poincare section are shown.

A modified radio­electronic analog of the nonlinear ring cavity is realized in laboratory. The device represents a special class of oscillations or waves sources. An operation principle of the sources is based on interference amplification of feedback signal by an input signal. A laboratory experiments are performed, the likeness of their results and simulation data is shown. An intermittency, chaos, regular, static modes are detected. A thesis on controlled nonlinearity of dynamical systems is suggested.

The problem of detection and quantitative characterization of nonlinear directional couplings between stochastic oscillators is considered. Coupling characteristics and a technique for their estimation from time series are suggested. An analytic expression for a statistical significance level of the conclusion about coupling presence is derived that allows a reliable inference from relatively short signals. Performance of the approach is demonstrated in numerical experiments with diverse individual properties of oscillators and different kinds of coupling functions.

In the present work the new approach to numerical research of the concrete multi­dimensional and multiparametric dynamic systems is submitted. The offered approach, in part realized and approved, is based on computer calculation of phase trajectories and on use of pattern recognition methods.