Sergey Kashchenko

Spatially distributed integro-differential systems of equations with periodic boundary conditions are considered. In applications, such systems arise as limiting ones for some nonlinear fully coupled ensembles. The simplest critical cases of zero and purely imaginary eigenvalues in the problem of stability of the zero equilibrium state are considered.
In these two situations, quasinormal forms are constructed, for which the question of the existence of piecewise constant solutions is studied. In the case of a simple zero root, the conditions for the stability of these solutions are determined. The existence of piecewise constant solutions with more than one discontinuity point is shown. An algorithm for calculating solutions of the corresponding boundary value problem by numerical methods is presented. A numerical experiment is performed, confirming the analytical constructions.

This paper studies the dynamics of a system of two coupled self-excited oscillators of first order with on-off delayed feedback using numerical and analytical methods. Regions of “fast” and “long” synchronization are identified in the parameter space, and the problem of synchronization on an unstable cycle is examined. For small coupling coefficients it is shown by analytical methods that the dynamics of the initial system is determined by the dynamics of a special one-dimensional map.

We consider the local dynamics (in a neighborhood of the equilibrium state) of nonperiodic chains of $N$ identical elements, each described by a second-order equation. It is assumed that the coupling between chain elements is one-sided. The main assumption is that the number $N$ of elements is sufficiently large, i. e., the small parameter is $N^{−1}$. Critical cases in the stability problem for the zero solution are identified, and it is shown that they have infinite dimension in the sense that infinitely many roots of the corresponding characteristic equation tend to zero as the small parameter $N^{−1}$ tends to zero. Known methods of local analysis based on the use of the method of invariant manifolds and the method of normal forms are not directly applicable in the problems under consideration. The main results consist in constructing so-called quasinormal forms — families of special nonlinear boundary value problems of parabolic type, which play the role of classical normal forms. Bifurcation phenomena are investigated and asymptotics of families of solutions bifurcating from the equilibrium state are constructed. The coupling parameter between chain elements and the parameter appearing in the boundary condition turn out to be fundamentally important. It is shown that the structure of solutions for nonperiodic chains is generally more complex than for periodic ones (ring chains).