Прикладные задачи нелинейной теории колебаний и волн

Topic. The work is devoted to the study of a mathematical model describing an optical system with two-dimensional feedback. An example of such an optical system could be a nonlinear interferometer with a specular reflection of the field. The mathematical model is a nonlinear functional differential parabolic equation with the transformation of the spatial variable reflection and conditions on the disk. Aim of the work is to study the conditions for the occurrence of spatially inhomogeneous stationary solutions.

Issue. The article is devoted to a numerical study of the dynamics and advection in a vortex parquet. A vortex structure, consisting of vortex patches on the entire plane, is considered. The mathematical model is formulated as a system of two partial differential equations in terms of vorticity and stream function. The dynamics of the vortex structures is considered in a rectangular area under the assumption that periodic boundary conditions are imposed on the stream function. Investigation methods.

Topic and aim. In this work, we study the dynamics of the kicked van der Pol oscillator with the amplitude of kicks depending nonlinearly on the dynamic variable. We choose the expansions of the function cos x in a Taylor series near zero, as functions describing this dependence.

Aim. The aim of the work is the numerical study of the oscillations of two-phase medium (a mixture of gas and a dispersed phase of solid particles) which caused by electric charge of the dispersed component, and the reciprocal effect of the dynamics of gas and solid particles, as well as the effect of linear size of dispersed particles on the dynamic processes. Methods. With the help of numerical models of electrically charged suspension was modeled in different modes of oscillatory dynamics in a dusty environment.

Theme. The dynamics of a nonlinear numerical model of a nonlinear optical interaction in the semiconductor disk laser resonator under influence of the time delay is investigated. The conditions of self-excitation, stationary generation modes and their stability are studied. Methods. The analysis of stationary generation stability was performed with DDE-Biftool package. Analysis of higher dimensional regimes was performed using numerical integration, construction of phase portraits, spectra and calculation of Lyapunov exponents. Results.

Topic. We study the interaction of two fundamentally different synchronization mechanisms: by means of coupling and by means of the driving by a random signal, which is identical for all oscillators – common noise. Special attention is focused on the effect of frequency divergence arising from the competition between these mechanisms. Aim. The aim of the paper is to construct a universal theory describing such an interaction for a general class of smooth limit-cycle oscillators with a global coupling.

Issue. The paper investigates the behavior of solutions of a logistic equation with two delays from some neighborhood of the equilibrium state with a large value of the coefficient of linear growth. Such problems arise in modeling the population size taking into account the age structure, as a model of the number of insets, etc. Innovation. It is shown that the critical cases arising in the problem of the stability of an equilibrium state have infinite dimension: an infinitely large number of roots of the characteristic equation tend to the imaginary axis.

Aim. Construction a model of infection spread in the form of a lattice of stochastic cellular automata which can demonstrate nontrivial oscillating regimes; investigation of its dynamics and comparison with the mean-field model. Method. Numerical simulation of the square lattice of cellular automata by the Monte Carlo approach, theoretical and numerical study of the structure of the phase space of its mean-field model. Results. A modified SIRS-model of epidemic propagation has been proposed in the form of a lattice of stochastic cellular automata.