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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.

Currently, the method of nonlinear Granger causality is actively used in many applications in medicine, biology, physics, to identify the coupling between objects from the records of their oscillations (time series) using forecasting models. In this paper the impact of choosing the model structure on the method performance is investigated. The possibility of obtaining reliable estimates of coupling is numerically demonstrated, even if the structure of the constructed forecasting model differs from that of the reference system.

In the Pliocene (about ?ve – two million years before present) global climate ?uctuated with a period corresponding well 41-thousand-year cycle of changes in the Earth’s axis inclination to the ecliptic plane. Then, this period has disappeared, despite the fact that the 41-thousand-year cycle even slightly increased its scope and, therefore, the response to it would have only strengthened. By analyzing paleoclimatic series covering the Pliocene and subsequent Pleistocene, we show that the response of the climate system simply became unstable and therefore unobservable.

In the work the behavior of a van der Pol oscillator with a hard excitation is considered near the excitation threshold under parametrical (multiplicative) Gaussian white noise disturbances, and in a case of the two noise sources presence: parametrical one and additive noise. The evolution of probability distribution is studied when a control parameter and a noise intensity are changed. A comparison of the theoretical results, obtained in the quasi-harmonic approach with the results  of numerical solutions of the oscillator stochastic equations is ful?lled.