Applied Problems of Nonlinear Oscillation and Wave Theory

Characteristics of modes of the planar waveguide with thin walls, which are made from nonlinear media, are studied. The effect of the conversion of a mode from one set to another is considered. This effect may occur if parameters of the waveguide or transmitted power are varied. It is shown that the effect of the guided or leaky modes «disappearance» can arise when the field amplitudes increase, i.e., high powers launched into such a structure can suppress the modal propagation.

We consider the population dynamics model «predator – two preys». A deterministic stability of limit cycles of this three­dimensional model in a period doubling bifurcations zone at the transition from an order to chaos is investigated. Stochastic sensitivity of cycles for additive and parametrical random disturbances is analyzed with the help of stochastic sensitivity function technique. Thin effects of stochastic influences are demonstrated. Growth of stochastic sensitivity of cycles for period doubling under transition from order to chaos is shown.

We investigated coupled map lattices based on the Ricker model that describes the spatial dynamics of heterogeneous populations represented by two connected groups of individuals with a migration interaction between them. Bifurcation mechanisms in­phase and antiphase synchronization of multistability regimes were considered in such systems. To identify a synchronization mode we introduced the quantitative measure of synchronization.

Based on the wavelet­analysis of experimental data, we study in this paper the phenomenon of synchronization of oscillations in the dynamics of small groups of structural units of the kidney (paired nephrons and triplets). Distinctions between synchronous dynamics of interacting nephrons in normal and hypertensive rats are discussed. We show that mean duration of synchronous oscillations is about 3 times less in hypertensive rats. We state that in­phase synchronization is the most typical case in the dynamics of interacting nephrons (more than 90% of experimental data).

Estimating coupling between systems of different nature is an urgent field of nonlinear dynamics method application. This work aims to compare classical linear Granger approach and its nonlinear analogues based on analysis of ethalon dynamical systems and neurophysiological data. The results achieved show nonlinear approach to be more sensitive, and so it is able to detect significant coupling, when linear one fails.

Nonlinear dynamics of the ensemble consisting of two phase­locked generators, which are coupled in a ring with feedback, is discovered. The conditions of stability of the synchronous regimes and appropriatenesses of excitation and progress of the non­synchronous regimes are examined within the bounds of the dynamic model with one and a half degrees of freedom. The extensive image of the dynamic regimes and bifurcating transitions, creating resources for the formation in the system of various types of oscillations, is discovered.

The analysis of heart rhythms using symbolic dynamics is perfomed. Time intervals corresponding to the predominance of sympathetic or parasym­pathetic tone of the nervous regulation are encoded. During encoding 25 symbols are used, what leads to a wide variety of words in the symbolic strings. The analysis of heart rhythms for patients of all ages, including healthy ones and patients with cardiovascular diseases are produced. These results give characteristic of age­related changes and different pathologies in cardiac rhythms.

A symmetrical two­dimensional flow past two rotating circular cylinders in a side­by­side arrangement is numerically investigated. Each cylinder is partially covered with an impermeable shroud in such a way that the unshielded moving section faces the incident flow. The effect of flow speed and tangential speed of the cylinder surface on flow topology is investigated for Reynolds numbers from 0 to 100. The formation of stationary eddies – «turrons» – in front of the gap between the cylinders is shown for a wide range of governing parameters.

It is offered to consider Sedov’s self­similar solution which earlier was used for exposition only an initial stage of flow from strong blast, in a role of an intermediate asymptotics of matching flow and for any medial, but not so major moment of time. Thus the index of self­similarity should be increased. The upper border of this range is certain from a condition of a constancy of an entropy behind a shock wave.

Synchronization phenomena are studied in phase dynamics approximation in the periodically driven system of two coupled oscillators. The cases are discussed when the autonomous oscillators demonstrate phase locking or beats with incommensurate frequencies. Lyapunov charts are presented, the possible regimes of dynamics of the driven system are discussed. Different types of two­dimensional tori are revealed and classified.