-

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.

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.

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.

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.

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.

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.

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.

It was shown that addition of modulation of driving parameter with using delay can be considered as physically reasoned method of construction two-dimensional maps with nonfixed Jacobian. The examples of such two-parameter and three-parameter maps were presented. The conditions of Neumark–Sacker’s bifurcation, period doubling and resonance 1:2 were obtained. The structure of parameter space was studied by dynamical regimes maps method and the regions of quasiperiodic regimes and different synchronous regimes were revealed.

Characteristics of the leaky modes, propagating along the planar layered waveguides with nonlinear media, are studied. The mode phase constants and attenuation coefficients are calculated. In nonlinear structures the dependencies of the mode field distributions on the longitudinal coordinate are shown to differ from exponential ones which are typical for the linear problems. This property results in the effect of the different modes transformation even for the regular geometry of the waveguide.

In this paper we studied intermittent modes in the two-parametric set of onedimensional maps with the neutral unstable point at a phase space boundary. We built the phase diagram in a space of parameters. It defines possible transitions to chaos with a parameter change. We showed the unusual mode of the intermittency concurrence. We studied the laminar length distribution function, Lyapunov exponent and topological entropy of this maps set.