chaos.

Realization of stable two­frequency oscillations is shown in the Maxwell–Bloch model. Birth of a stable ergodic two­dimensional torus from the critical closed curve is observed. The conditions of the passage to chaos via a cascade of torus doubling bifurcations are obtained. It is established that at bifurcations points a structurally unstable three­dimensional torus is produced, which gives rise to a stable doubled ergodic torus. Analytical approximation describing dynamics of the system near a point of torus birth is found.

The dynamics of an astrocyte model is investigated. The astrocytes represent a type of glial cells regulating oscillations of major signaling cells, e.g. neurons. Subserved by complex molecular mechanisms the astrocytes generate calcium auto-oscillations which, in turn, are associated with the release of neuroactive chemicals into extracellular space. At variance with classical astrocyte models the three-component model considered takes into account a regulation of calcium release due to nonlinear dynamics of inositol-1,4,5 trisphosphate (IP3).

The model of two field­coupled spin­transfer oscillators has been derived and studied. It has been shown that this model demonstrates phase synchronization in a wide bandwidth, quasiperiodic oscillations and chaos.

Regularities of multistability developments are considered in an array of identical self­sustained oscillators with transition to chaos through period­doubling bifurcations. The used model is chain of diffusivelly coupled Rossler oscillators. The number of coexisting regimes are determined through the cascade of the bifurcations. It is shown that regularities of incresing of attractors are defined be transformation of the phase spectrum duing transition to chaos.

The problem of the dynamics of three coupled self­oscillators and three coupled periodically driven self­oscillators is discussed, in the last case only one of the oscillators is directly exited by the external fore. The regions of complete synchronization, two­, three­and four­frequency tori and chaos are revealed. Three typical situations of synchronization of three self­oscillators by the external driving are found.

In this paper we research a mathematical model of dynamics for the population number. We considered the population of the two­age classes by the beginning of the next season: the younger, one including not reproductive individuals, and the senior class, consisting of the individuals participating in reproduction. The model parameters (birth rate and survival rates) represent the exponential functions of the both age groups numbers. According to this supposition the density­dependent factors restrict the development of population.

We review the results of studies of spontaneous and stimulated emission of relativistic particles in natural and photonic crystals.We consider the diffraction of electromagnetic waves in a crystal, and the resonance and parametric (quasi-Cherenkov) X-ray radiation, the radiation in the channeling of relativistic particles in crystals, diffraction radiation in conditions of channeling, diffraction radiation of a relativistic oscillator, induced radiation in multidimensional space-periodic resonators (natural or artificial (electromagnetic, photonic) crystals).

These lectures were delivered to the high school students at the School – seminar «Nonlinear Days for Youth in Saratov – 2012» in October 2012. They present the two ways of historical investigation of symmetry. The first way is self-similarity, i.e. invariance at dimension scale changing. In a more general way the term «scaling» is used, meaning the existence of power-law correlation between some variable and variables x1, ...xn: y = Axα1...xαn1 (self-similarity) appearing in various fields of science and culture. G.I. Barenblatt indicates that scaling laws never appear by accident.