Asymptotic behavior of singularly perturbed systems with periodic nonlinearities and external forces

Abstract

In this paper we consider singularly perturbed phase synchronization systems with external disturbances. The systems are described by integro-differential Volterra equations with periodic nonlinear functions and a small parameter at the higher derivative. The disturbed systems examined in this paper have (like undisturbed ones) infinite sequence of equilibrium points. So for them the main problem of phase synchronization systems remains: whether the system is gradient-like, i.e. its any solution converges to one of equilibria. In this paper we offer frequency-algebraic criteria which guarantee that the convergence of any solution of undisturbed system under singular perturbation is not destroyed by external disturbance. If the system is not gradient-like it may have periodic solutions. We demonstrate that the relaxation of frequency-algebraic criteria leads to conditions for the absence of high frequency periodic solutions. The results of the investigation are uniform with respect to the small parameter.