Dynamics of a System of Two Simplest Oscillators with Finite Non-linear Feedbacks - A. A. Kashchenko (Моделирование и анализ информационных систем)
In this paper, we consider a singularly perturbed system of two diﬀerential equations with delay which simulates two coupled oscillators with nonlinear feedback. Feedback function is assumed to be ﬁnite, piecewise continuous, and with a constant sign. In this paper, we prove the existence of relaxation periodic solutions and make conclusion about their stability. With the help of the special method of a large parameter we construct asymptotics of the solutions with the initial conditions of a certain class. On this asymptotics we build a special mapping, which in the main describes the dynamics of the original model. It is shown that the dynamics changes signiﬁcantly with the decreasing of coupling coeﬃcient: we have a stable homogeneous periodic solution if the coupling coeﬃcient is of unity order, and with decreasing the coupling coeﬃcient the dynamics become more complex, and it is described by a special mapping. It was shown that for small values of the coupling under certain values of the parameters several diﬀerent stable relaxation periodic regimes coexist in the original problem.