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A new model of neurons like elements is suggested in the paper. The model is based on the generalized Lottka–Volterra model with excitatory coupling. The study is motivated by the fact that the excitatory couplings are the dominating type of interactions between neurons in the human brain. It is shown in the paper that there are two regimes exist in such ensemble of oscillators in dependence on the coupling between the elements: the regime with stable heteroclinical cycle and the regime with stable limit cycle.

A short review of contribution to the deterministic chaos theory, that had been made by professor Alexander F. Goloubentsev (Saratov University), is given.

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 ?rst 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 = Ax?1...x?n1 (self-similarity) appearing in various ?elds of science and culture. G.I.

This article is an extended summary of a lecture given to undergraduate and postgraduate students of Saratov State University’s Faculty of Nonlinear Processes at the School-cum-Conference «Nonlinear Days for the Young in Saratov – 2012».

Built in a cell synthetic gene regulatory elements may function rather independently on the original natural system. Experimental and theoretical studies of small synthetic networks allow for a better understanding of fundamental dynamical mechanisms of gene regulation. This paper gives an introduction to the modern mathematical approaches and methods in this ?eld, primarily in the framework of nonlinear dynamics.

The regime of the simultaneous existence of space and surface waves in the O-type oscillator with a periodically inhomogeneous comb has been considered. The impedance and quality factor dependences for space waves, surface waves and hybrid mode have been compared.

We explore the destruction of phase multistability which takes place in an ensemble of period doubling oscillators under the action of long-distance couplings, which appear randomly between the arbitrary cells. The investigation is carried out on the example of a chain of Rossler’s oscillators with periodic boundary conditions, where alongside with local couplings between the elements exist long-range interconnections. The sequence of bifurcations, which accompany increasing of the strength of the global coupling is determined.

The problems of existence and stability of the symmetry-induced nonlinear normal modes in the electric chain of non-linear capacitors, connected to each other with linear inductors (the model described in Physica D238 (2009) 1228) are investigated. For all modes of this type, the upper limit of the stability region (in amplitude of voltage oscillations on capacitors) as a function of the chain cell number were found. Asymptotic formulas were determined at cell number tends to in?nity.

The task of detection statistically signi?cant interaction, its direction and delay between time data series of two oscillatory systems in case of close coupling is investigated with nonlinear modeling approach. Numerical experiments on oscillatory model systems with di?erent coupling function variants are used to study main dependences.

The problem of detection and quantitative estimation of directional couplings (mutual in?uences) between systems from discrete records of their oscillations (time series) arises in di?erent ?elds of research. This work shows that results of the traditional «Granger causality» approach depend essentially on a sampling interval (a time step). We have revealed the causes and character of the in?uence of a sampling interval on numerical values of coupling estimates.