Applied Problems of Nonlinear Oscillation and Wave Theory

We consider a mechanical system consisting of two controlled masses that are attached to a movable platform via springs. We assume that at the absence of interaction the oscillations of both masses are described by the van der Pol equations. In this case, different modes of synchronous behavior of the masses are observed: in-phase (complete), anti-phase and phase locking. By the methods of qualitative and numerical analysis, the boundaries of the stability domains of these regimes are obtained.

In this paper the intermittent behavior taking place near the boundary of the synchronous time scales of interacted chaotic oscillators being in the synchronous regime is studied. At the regime of time-scale synchronization the system demonstrates synchronous dynamics in a certain range of the time scales whereas the processes on the other time scales remain asynchronous. On the basis of analysis of statistical characteristics of the intermittent behavior, i.e.

Effect of noise on generalized synchronization in spatially extended systems described by Ginzburg–Landau equations being in the spatio-temporal chaotic regime is studied. It is shown, that noise does not affect the synchronous regime threshold in such systems. The reasons of the revealed particularity have been explained by means of the modified system approach and confirmed by the results of numerical simulation.

From the viewpoint of the order parameters concept spatial pattern formation in excitable fluctuating medium was researched analytically. The reaction–diffusion system in external noise was considered as a model of such medium. Stochastic equations for the unstable modes amplitudes (order parameters) and the dispersion equations for the average unstable modes amplitudes were received. Fokker–Planck equation for the order parameters was received.

In paper on basis of radiophysical experiment analysis of dynamics of the FitzHugh–Nagumo system have been carried out. The dependence of oscillation’s regime in the system from force parameter has been found out. Influence of the form of the external force signal on the system response has been studied.

Some problems of modelling of dynamic systems with piecewise linear characteristics are considered. New methods of approximation of the piecewise linear functions, in particular, step functions without disadvantages of the traditional Fourier series expansions are suggested. Some questions of convergence and error estimation of the approximation are explored.

This work presents experimental results of synchronization of klystron oscillator with delayed feedback in the presence of noise. It was shown, that synchronization effect significantly decreases noise level, leads to amplifying of synchronizing signal and shortening of synchronization wideband. When additive sum of harmonic and noise signal are directed to klystron input, signal-to-noise ratio significantly increases due to synchronization.

The methods are proposed for the reconstruction of time-delay systems modeled by neutral  delay-differential equations from their time series. The methods are successfully applied to the recovery of generalized Mackey–Glass equation and equations modeling ship rolling and human movement from simulated data.

The modes of genetic structure and size dynamics of structured population are investigated in this work. The reproductive potential and survival rate of reproductive part of population in following years of life are determined on genetic level. It has been shown that evolutional increasing of average population fitness is followed by arising of complicated dynamics of population size and of genetic structure. Further growth of fitness is capable to stabilize the genetic structure of population and so only the population size will be fluctuating with regular or chaotic circling.

We propose a new method of control of phase multistability in two coupled self­sustained oscillators. The method is based on the «pulling» of phases of oscillations to the target mode under two external harmonic forces, which influence the first and the second sub­systems simultaneosly. Varying the phase shift between the external signals results in control of switching between coexisting oscillating modes. Effectiveness of the method is demonstrated on the example of switching between periodic and chaotic regimes in two Chua’s oscillatotrs.