# Журнал в журнале

## Billiard type systems and Fermi acceleration

Systems of billiard types with perturbed boundaries are described. A generalized dispersing billiard – the Lorentz gas with the open horizon – and a focusing billiard in the form of stadium are considered. It is analytically and numerically shown that, if the billiard possesses the property of the developed chaos, the consequence of the boundary perturbation is the Fermi acceleration. However, the perturbation of the nearly integrable billiard system leads to a new interesting phenomenon – the separation of the billiard particles in their velocities.

## 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 field. Two different types of virtual cathode dynamics were discovered. The value of external magnetic field determines a dominant type of dynamics. Therefore the current critical value (when virtual cathode arises in a beam) depends on external magnetic field 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.

## 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 different 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 differential 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.

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

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

## Coupled van der pol and van der Pol–Duffing oscillators: dynamics of phase and computer simulation

Synchronization in the system of coupled nonidentical and nonisochronous van der Pol oscillators with dissipative and inertial type of coupling is discussed. Generalized Adler equation is obtained and investigated in the presence of all factors. Basic symmetry of the equation, with leads to equivalence of some physical factors, is displayed. Numerical investigation of parameters space of initial differential system is realized. Results of two methods are compared and 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 affect the firing patterns of the neighboring neurons. To take into account mentioned above effects, 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 effects being the attributes of ionic coupling that are include the emergence of new time scales and spatially-ordered firing patterns.

## Estimation of mixing velocity in chaotic systems

In the paper an effect 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.

## 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 find that the periodic force with a certain frequency applied to one of the oscillators enables to switch the ring to another stable regime.

## Synchronization of two coupled klystron active oscillators with delayed feedback

Results of experimental research of synchronization of two coupled almost identical resonance microwave active oscillators on multicavity klystrons in the modes of periodic and chaotic oscillations are presented. It is shown that depending on type of coupling it is possible to realize a mode of mutual frequency capture, synchronization by means of chaos full elimination by outer harmonic signal, and full synchronization mode. A possibility of using the chaos elimination effect for generation of sequence of chaotic radio pulses is shown.