# Журнал в журнале

## Autooscillating system with compensated dissipation: dynamics of approximated discrete map

The pulse-driven van der Pol oscillator with the external pulse amplitude depending on the system variables is considered. The discrete map for values of the system variables just before the pulse moment was obtained by the slow-varying-amplitude method. Further the parameter space of this map was analyzed, and the existence of the Hamiltonian critical behavior in this system was shown. The remarkable fact is that our system is the system with the dissipation depending not only on the parameter values, but on the variable values too.

## Synchronization of spatial-periodic modes in the ring of oscillators with phase multystability

We study external synchronization of periodic oscillations in a ring of oscillators driven by periodic force. It is shown that each multistable state that co-exists in the system possesses its own synchronization region. We ?nd that the periodic force with a certain frequency applied to one of the oscillators enables to switch the ring to another stable regime.

## Estimation of mixing velocity in chaotic systems

In the paper an e?ect of phase space trajectories mixing in chaotic systems is considered. Approximate analytic estimations are given of mixing dynamics in discrete and continuous chaotic systems. Easy algorithm is developed for experimental calculation of mixing degree and mixing velocity, both local and average over the attractor. Results of this algorithm application to Henon map and to Chua system are discussed.

## Noise-induced coherent firing patterns in small neural ensembles with ionic coupling

By means of modeling and numeric simulation we consider, how the rise of extracellular potassium concentration due to the neuronal activity can a?ect the ?ring patterns of the neighboring neurons. To take into account mentioned above e?ects, we suggest simple extension of Hodgkin-Huxley model. We consider the behavior of 2, 4, and 8 excitable neurons being forced by external noisy stimulus. We reveal the main e?ects being the attributes of ionic coupling that are include the emergence of new time scales and spatially-ordered ?ring patterns.

## Asymptotics of complex spatio-temporal structures in the systems with large delay

The local dynamics is considered of di?erential equations with two delays in the case of one delay is asymptotically large. Under this condition, critical cases have in?nite dimension. As the normal form equations the Ginzburg–Landau equations have been. Their nonlocal dynamics de?nes local behavior of solutions of initial equations.

## Group-theoretical methods for simplification of stability analysis of dynamical regimes in nonlinear systems with discrete symmetry

We present a detailed description of the group-theoretical method which has been published in 2006 by the authors. This method can frequently simplify the study of the stability of di?erent dynamical regimes in nonlinear physical systems with discrete symmetry since it allows one to split the set of the linearized (near a considered regime) nonlinear di?erential equations into a number of independent subsets of small dimensions. The above method is illustrated with the case of stability analysis of some dynamical regimes in the simple octahedral structure.

## External magnetic field influence on the forming and dynamics of virtual cathode

The results of investigation of virtual cathode mechanisms forming and its dynamics in the context of 2-dimensional model are presented. There were considered entire and tubular electron beams in external axial magnetic ?eld. Two di?erent types of virtual cathode dynamics were discovered. The value of external magnetic ?eld determines a dominant type of dynamics. Therefore the current critical value (when virtual cathode arises in a beam) depends on external magnetic ?eld value.

## Qualitative analysis of one class of optoelectronic systems singularly perturbed models

Two models of semiconductor laser with delayed optical feedback are studied. We consider singularly perturbed problem because of the large parameter presence. We construct and discuss quasinormal forms of models in trancritical cases.

## Synchronization of two-frequency quasi-periodic oscillations

In present paper we study the e?ect of synchronization of two-frequency quasiperiodic oscillations. We analyze both external and mutual synchronization. The peculiarities of synchronization of a resonant limit cycle on a two-dimensional torus are established. It is shown that in general case, one and then another one of the basic frequencies is locked. The results of computer simulation are con?rmed experimentally.

## Peculiarities of calculation of the lyapunov exponents set in distributed sele-oscillated sistems with delayed feedback

The numerical scheme for calculation the set of Lyapunov exponents in distributed systems with delayed feedback based on a modi?cation of Benettine algorithm is described. The results of numerical simulation of two such systems (active oscillator with cubic nonlinearity and active oscillator of klystron type) are presented. The sets of Lyapunov exponents in di?erent regimes, particularly in regimes of «weak» and «developed» chaos are analyzed. The calculation peculiarities of the set of Lyapunov exponents in the systems with delayed feedback are discussed.