Applied Problems of Nonlinear Oscillation and Wave Theory

In the work nonlinear Duffing oscillator is considered under impulse excitation with two ways of introduction of the random additive term simulating noise, - with help of amplitude modulation and modulation of period of impulses sequence. The scaling properties both in the Feigenbaum scenario and in the tricritical case are shown.

Mathematical model of ion fluxes across the cell membrane of algae Chara coralline is developed. The transient processes and self-oscillating modes connected with potential-dependent transport of protons across the membrane are considered. Important role ofthese processes for plant cell is discussed.

This paper demonstrates a tool for prediction time series on a base of iterated function system of the theory of fractals. Iterations result in an attractor or fractal in a space of compacts. The attractor is a support of invariant probabilistic measure or multifractal in a space of Borel measures. An inverse problem consists of finding iterated function system and its probabilities by means of empirical measure. The estimates might be obtained from time series by symbolic dynamics methods.

We present the analysis of spatiotemporal dynamics in the system modeling collective behaviour of ensemble of electrically coupled neuronal cells. The dynamics of local element is described by the FitzHugh – Nagumo system with complex threshold excitation. Heteroclinic orbits and corresponding wave fronts are investigated. We show that in the phase space of system for traveling waves there exist heteroclinic cycle formed by separatrix manifolds of two saddle-foci.

In this paper we study the phenomena of interaction between three rhythmic components of renal autoregulation. Clear distinctions of the corresponding phenomena for almost periodic dynamics of nephrons observed in normotensive rats and irregular (chaotic) dynamics that occur in the nephrons of genetically hypertensive rats are revealed.

The complex dynamics and the peculiarities of the transition to chaos in two coupled flow systems – Chua’s circuits are investigated. It is shown that this system demonstrates more complicated behavior at the onset of chaos than the discrete maps. In particular, the codimension of the critical behavior increases in such system.

The particular properties of dynamics are discussed for the dissipatively coupled van der Pol oscillators, nonidentical in values of parameters controlling the Hopf bifurcation. The opportunity of a special synchronization regime in an infinitively long band between oscillation death and quasiperiodicity areas is shown for such system.

A mechanism of collective generation of bursts in ensembles of spiking neurons with nonlocal excitatory coupling is studied. Three types of the network topology is considered: (a) chains with regular short-range nonlocal coupling, (b) chains with a small number of random long-range connections and dominating regular short-range ones, (c) random ensembles with a power law of node degree distribution. It is shown, that there exists a common mechanism of burst generation resulting from instability of synchronous slow spiking as the coupling strengthens, giving rise to fast repetitive spikes.

It was shown experimentally that at synchronization of chaotic oscillation in the microwave active resonance oscillators the following effects take place: own chaotic dynamics suppression; periodic oscillation establishing; the frequency capture by external harmonic signal, and noticeable power decrease of these oscillation.

We present the results of investigation of dynamical regimes in the models of oscillatory systems with frequency and frequency-phase control. The processes of excitement of nonsynchronous regimes and transitions between them are considered. A special attention is given to the study of the origin and stages of development of deterministic chaos in these systems. The existence of several types of chaotic attractors is established. Various scenarios of transition from regular dynamical regimes to chaotic ones under variation of the systems parameters are analyzed.