Dynamics of a System of Two Simplest Oscillators with Compactly Supported Nonlinear Feedbacks

Abstract

In this paper, we consider a singularly perturbed system of two differential equations with delay, simulating two coupled oscillators with a nonlinear feedback. The feedback function is assumed to be compactly supported and piecewise-continuous and it is assumed that its sign is constant. In this paper, we prove the existence of relaxation periodic solutions and make conclusions about their stability. Using a special large-parameter method, we construct asymptotics of all solutions of the considered system under the assumption that the initial-value conditions belong to a certain class. Using this asymptotics, we construct a special mapping principally describing the dynamics of the original model. It is shown that the dynamics changes fundamentally as the coupling coefficient decreases: we have a stable homogeneous periodic solution if the coupling coefficient is on the order of unity and the dynamics become more complex as the coupling coefficient decreases (it is described by a special map). For small values of the coupling, we show that there are values of the parameters such that several different stable relaxation periodic regimes coexist in the original problem.