# Дебют

## Elementary theory of interaction between space charge waves and electromagnetic waves for the transverse-current traveling-wave tube

Analytical theory for the transverse-current traveling-wave tube has been created with the use of successive approximations method. Interaction between space charge waves and electromagnetic waves has been examined. Numerical calculations have shown that the inclusion of the backward wave significantly changes the pattern of interchange of energy – there is a shift of zero values of reactive interaction power by π/3.

## About approximate analytical solutions of Lotka–Volterra equations

The possibility of analytical decision of Lotka–Volterra equation is demonstrated for «predator–prey» model, and for comlicated models.

## INFLUENCE BOTH OF REFLECTIONS AND DISSIPATION IN THE BACKWARD-WAVE OSCILLATOR ON FIRST RESONANCE PEAK OF AMPLITUDE AT THE TRANSIENT BEGINNING

In the present work an effort to analyze dissipation and the wave reflection influence on the first resonant pulse in the time dependence of the output signal of the BWO has been taken.

## Radial patterns in a vibrated granular layer

Laboratory experiments were conducted for a sand layer placed in the verticallyoscillated containers of various shapes. Radial patterns on the sand surface were observed; experimental investigations of such structures have never been described in scientific literature. The waveform, amplitude and frequency of vibrations and the depth of the vibrated layer could be varied, allowing study the dependence of the shape and scale of radial structures upon these parameters.

## Evolutionary stochastic model of dynamics of psycho-emotional states in affective disorders

Experimental researches in the field of psychiatry show that variation of psychoemotional states in humans resembles switching process between the various major phases (depressed, normal, manic state). The parameters of such switching process significantly differ in normal state and in affective disorders. To date, the known mathematical models of this process, are based on the assumption of presence of internal chaotic dynamics.

## Dispersion equation for various models of Pierce diode

Peculiarities of derivation of a dispersion equation for various models of Pierce diode have been sequentially considered in this paper.

## Solitary waves of two-dimensional modified Kawahara equation

Equations of this type describe a number of real-life processes like wave motion under ice mantle or propagation of waves of longitudinal deformation in thin cylinder shell. Using «Simplest Equation Method» exact solitary-wave solutions of the two-dimensional Kawahara Equation were obtained. On the basis of implicit pseudospectral method the numerical investigation is carried out. Regimes of two-dimensional deformation waves with classic solitary behavior were discovered.

## On two-dimensional linear theory of interaction between electron beam and traveling electromagnetic wave: allowing for influence of space charge in a thin beam model

In the article two-dimensional model of interaction between infinitely thin electron beam in longitudinal magnetic field and traveling electromagnetic wave has been considered; in the frames of two-dimensional linear theory integral equation described such interaction has been formulated. On the basis of derived dispersion relation condition of initiation of beam instability has been found and influence of space charge fields on the processes of interaction has been analyzed.

## Circular non-autonomous generator of hyperbolic chaos

A scheme of circular system is introduced, which is supposed to generate hyperbolic chaos. Its operation is based on doubling of phase on each complete cycle of the signal transmission through the feedback ring. That is a criterion for the attractor of Smale– Williams type to exist. Mathematically, the model is described by the fourth order nonautonomous system of ordinary differential equations. The equations for slowly varying complex amplitudes are derived, and the Poincare return map is obtained. Numerical simulation data are presented.