On Periodic Solutions of Singularly Perturbed Integro-differential Volterra Equations with Periodic Nonlinearities*

We examine nonlinear singularly perturbed systems, described by integro-differential equations with periodic nonlinearities. Equations with periodic nonlinearities govern phase-locked loops and other synchronization circuits, as well as many "pendulum-like" systems, arising in mechanics and physics. The presence of periodic nonlinearity typically endows the system with infinite sequence of equilibria points. One of the central questions related to such systems is whether any solution converges to one of the equilibria (which is sometimes referred to as the gradient-like behavior) or some oscillatory solutions exist. Under singular perturbation, the self-standing problem is the persistence of the gradient-like behavior as the small parameter tends to zero. In spite of substantial efforts in solving these problem, the existing conditions for the gradient-like behavior (which guarantee, in particular, the absence of oscillations) are only sufficient and may be quite conservative. In this paper we demonstrate that their relaxation guarantees inexistence of special oscillatory trajectories, namely, periodic solutions of high frequency. We give constructive frequency-domain conditions, which guarantee that all periodic solutions in the system, if they exist, have frequencies lower than some predefined constant. An important property of this estimate is its uniformity with respect to the small parameter.