Applied Problems of Nonlinear Oscillation and Wave Theory

Different features of the pulsed synchronization of self-oscillatory systems are considered. Namely nonisochronous, stabilization of the unstable systems, synchronization of the coupled oscillators in the region of the «oscillatory death» and etc. Illustrations for the coupled nonisochronously oscillators and nonidentical (controlling parameter and nonlinear dissipation) oscillators are presented.

We present the analysis of spatiotemporal dynamics of two-dimensional ensemble of electrically coupled FitzHugh–Nagumo neurons with oscillatory threshold. We show that in this system spatially localized activity structures can be formed. Such structures propagate through the system without changing their shape and velocity. We demonstrate that there exist two types of the structures: single and bound states. General characteristics of localized structures such as regions of existence, geometrical sizes and velocity are investigated.

Power characteristics of low-power microwave chaotic generators are investigated. Power characteristics of several variants of transistor generators are discussed. Current and power consumption necessary for generation of chaos with preassigned characteristics is analyzed, relationships between supply voltage, current draw and output signal power are shown on example of one of the generators. SiGe monolithic IC generator is designed and power characteristics of the generator are estimated.

An important problem of the change in the effective parameters of averaged motions in nonlinear systems is described. This problem is known in physics for a long time. It is concerned with the derivation of bodies motion equations taking into account the collisions with the molecules of the surrounding gas. Many researchers believe implicitly that this problem is essential only for the transfer from microscopic equations to macroscopic ones. However this problem reveals often itself in the present-day macroscopic physics.

In the present paper we study an unstable nonlinear oscillator in which the growth of amplitude of oscillations is limited by noise influence. We calculate the characteristics of noise-stabilized fluctuations. It is shown when the noise intensity changes, the system can demonstrate different effects such as the suppression of exponential instability of trajectories and the narrowing of the spectrum of fluctuations.

Synchronization is studied in ensembles of locally dissipative coupled and conservative coupled weak-nonlinear van der Pol oscillators. In the chain of N elements not less than 2N¡1 different regimes of global synchronization are stable at the same values of parameters. Cluster synchronization is considered as well. Existing of multiple fronts of synchronization switching is shown. These fronts go one through another without of changing or reflections from free boundaries.

Dynamics of a delayed-feedback oscillator with cubic nonlinearity driven by an external harmonic signal is considered in a case when in the free-running oscillator periodic regime is realized. Resonance curves, i.e. amplitude–frequency responses of the oscillator are derived analytically. Stability conditions for synchronization regime are analyzed. Synchronization tongues on the driving amplitude – driving frequency parameter plane are presented. General differences from classical picture of synchronization of the systems with one degree of freedom are discussed.

Mean field approximation is extended to multicomponent stochastic reaction­diffusion systems. A multivariate nonlinear self­consistent Fokker–Planck equation defining the probability density of the state of the system, which describes a well­known model of autocatalytic chemical reaction (Brusselator) with spatially correlated multiplicative noise, is obtained. The evolution of probability density and statistical characteristics of the system in the region of Turing bifurcation are studied.

The characteristics of surface TM modes, guided by diffuse interface between two nonlinear media (metamaterial and conventional dielectric), are investigated. The field distributions and dependencies of propagation constants on parameters of the transition layer are calculated. Effects, caused by media nonlinearity, are studied.