Sergey Kashchenko

We study the local dynamics of chains of coupled nonlinear systems of secondorder ordinary differential equations of diffusion-difference type. The main assumption is that the number of elements of chains is large enough. This condition allows us to pass to the problem with a continuous spatial variable. Critical cases have been considered while studying the stability of the equilibrum state. It is shown that all these cases have infinite dimension. The research technique is based on the development and application of special methods for construction of normal forms. Among the main results of the paper, we include the creation of new nonlinear boundary value problems of parabolic type, whose nonlocal dynamics describes the local behavior of solutions of the original system.

The dynamics of second-order equations with nonlinear delayed feedback and a large coefficient of a delayed equation is investigated using asymptotic methods. Based on special methods of quasi-normal forms, a new construction is elaborated for obtaining the main terms of asymptotic expansions of asymptotic residual solutions. It is shown that the dynamical properties of the above equations are determined mostly by the behavior of the solutions of some special families of parabolic boundary value problems. A comparative analysis of the dynamics of equations with the delayed feedback of three types is carried out.

We derive the discrete maps to describe the dynamics of coupled laser diodes. The maps allow us to find analytically regions of parameters and initial conditions in the functional phase space that correspond to spiking with stable (or nearly stable) phase shift. The method developed is promising for further discussion of controlled switching between periodic states by an impulse injection signal.