Local and Nonlocal Cycles in a System with Delayed Feedback Having Compact Support

The purpose of this work is to study small oscillations and oscillations with an asymptotically large amplitude in nonlinear systems of two equations with delay, regularly depending on a small parameter. We assume that the nonlinearity is compactly supported, i. e., its action is carried out only in a certain finite region of phase space. Local oscillations are studied by classical methods of bifurcation theory, and the study of nonlocal dynamics is based on a special large-parameter method, which makes it possible to reduce the original problem to the analysis of a specially constructed finite-dimensional mapping. In all cases, algorithms for constructing the asymptotic behavior of solutions are developed. In the case of local analysis, normal forms are constructed that determine the dynamics of the original system in a neighborhood of the zero equilibrium state, the asymptotic behavior of the periodic solution is constructed, and the question of its stability is answered. In studying nonlocal solutions, one-dimensional mappings are constructed that make it possible to determine the behavior of solutions with an asymptotically large amplitude. Conditions for the existence of a periodic solution are found and its stability is investigated.