Applied Problems of Nonlinear Oscillation and Wave Theory

In present paper the phenomena of coherence resonance, mutual and external synchronization of noise-induced stochastic oscillations in FitzHugh–Nagumo system are studied by means of numerical and natural experiments. The properties of attractor in the system as well as energy exchange processes are analyzed. Self-sustained character of stochastic oscillations in non-autonomous FitzHugh–Nagumo system justified.

Problem of interaction between self­sustained oscillating systems of different nature is discussed by an example of coupled brusselator and van der Pol oscillator. Picture of leading oscillator changing with the growth of coupling parameter is shown. Areas of different types of dynamics are indicated in the parameter space. The case of essentially different eigenfrequencies is discussed.

Multistable regimes in asymmetrically coupled logistic maps are investigated. The evolution of the multistability regions in the parameter plane and the basins of coexisting attractors are revealed.

Recent experimental and theoretical studies indicate that slow brain rhythms are generated by simple inhibitory neural networks. Sequential switching of tonic spiking activity is a widespread phenomenon underlying such rhythms. In this paper, we analyze a minimal, reciprocally connected circuit of three spiking units in the cases of different excitability classes of models. It is shown that in both types arising of stable heteroclic contour produces sequentail activation and slow rhythm generation in neural microcircuit. Bifurcation of heteroclinic contour arising is investigated.

The methods for the reconstruction of model delay-differential equations for ensembles of coupled time-delay systems from their time series are proposed. The methods efficiency is illustrated using chaotic and periodic time series from chains of diffusively coupled model and experimental time-delay systems for the cases of unidirectional andmutual coupling.

Transfer entropy is widely used to detect the directed coupling in oscillatory systems from their observed time series. The systematic error is detected, while estimating transfer entropy between nonlinear systems with K-nearest neighbours method.

A model for competition of two different species is considered. It is assumed that each consumer specializes on one resource only. The resource uptake rates are held constant.The basic feature of the model is that the dynamics of the resource is much slower than that of the consumer.The two consumers are coupled through direct reciprocal inhibition. Besides, self-limitation of the consumers due to overcrowding is also taken into account. The resources are noninteractive.

The authors propose a complex approach for the analysis of composite materials, including the fundamental models of the nonlinear dynamics, model of effective medium and the theory of electrical circuits. The composite consisting of spherical inclusions in the matrix is considered. The simulation of composite material is carried out by various methods.

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Results of researching linear theory of two models of resistive wall amplifier M-type are presented.

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Non-linear nonstationary theory of two-stream amplifier has been suggested for case of co- and opposite directed beams. Non-linear theory has been compared with calculation of linear theory for both cases.