Applied Problems of Nonlinear Oscillation and Wave Theory

The simplest kinetic equation for description of the electron-positron plasma vacuum creation in a strong linearly polarized electric («laser») field was reduced to the nonlinear ordinary differential equation of the second order. The corresponding truncated equation without the dissipative contributions was obtained also. In area of the tunnel mechanism action the non-local under an external field solutions for the residual electron-positron plasma was first obtained.

In the paper one of ways of spinup of rotors in devices with noncontact suspension is considered. For the creation of rotation the pulse-position way of control the fields of stator’s coils is applied. The method of approximation of limiting cycles by the Fourier series is carried out for the theoretical investigation of an opportunity of realization of the suggested way. The algorithm of control the coils fields is found and the conditions imposed on the parameters of the working moments which allow to receive the maximal angular speed of the rotor at the minimal power inputs are received.

Robust stability conditions in terms of linear matrix inequalities for a parametrically disturbed pendulum are obtained. Numerical results for estimating radius of robust stability are given.

Article is devoted to the nonconventional approach to the description and properties studying of the output flows arising in cyclic systems of mass service. This approach with use of imitation method allows to solve Webster – Allsop problem about delays in cyclic systems of mass service.

This work presents a dynamical phenomenon strongly related with the problems of synchronization and control of chaotic dynamical systems. Considering externally driven homoclinic chaotic systems, it is shown experimentally and theoretically that they tend to synchronize with signals strongly correlated with the saddle cycles of their skeleton; furthermore, when they are perturbed with a generic signal, uncorrelated with their skeleton, their chaotic behavior is reinforced.

The methods of sensitivity analysis of cycles under deterministic and stochastic disturbances for Higgins model describing glycolytic self-oscillations are considered. Two approaches connected with local exponents and stochastic sensitivity function are compared. The most sensitive parts of cycles are discovered. It was found that some parts of cycle lose stochastic stability along with stability increasing of cycle as whole.

In this paper, a comparative study of methods for classification of neuronal action potentials is performed, namely, the standard Principal Component Analysis (PCA) and techniques based on the wavelet-transform. It is shown that there are at least two caseswhen the wavelet-based approaches have advantages: 1) the presence of a small-scale structure in the shapes of spikes, and 2) the presence of slow noise of high intensity. It is stated that the quality of spike-sorting can be increased by signal’s filtering.

The results of numerical simulation of oscillations in a simple model of interaction of waves with inertial nonlinearity applicable to study of phenomena in a backward wave oscillator are produced. Convenient numerical characteristics for identification of anharmonic oscillations – a decrement of the autocorrelation function and the characteristic correlation time – have been used. It has been found that both sophistication and ordering of an excited signal were possible under influence of additive noise.

The dynamics of a system with unstable limit cycle under the periodic sequence of delta-pulses 

is considered. It is shown, that stable quasi-periodic regimes and phase lock regimes 

(synchronization) are observed within a narrow range of parameters of the external action in the 

system with cubic nonlinearity. Influence of main system’s parameters to the stable quasi-

periodic regimes and phase lock regimes is investigated.

Analogy between an approximate version of period-doubling (and period N-tupling) renormalization group analysis in complex domain and the phase transition theory of Yang-Lee (based on consideration of formally complexified thermodynamic values) is discussed. It is shown that the Julia sets of the renormalization transformation correspond to the approximation of Mandelbrot set of the original map. New aspects of analogy between the theory of dynamical systems and the phase transition theory are uncovered.