Singular perturbations of Volterra equations with periodic nonlinearities: Stability and oscillatory properties

Singularly perturbed integro-differential Volterra equations with MIMO periodic nonlinearities are considered, which describe synchronization circuits (such as phase- and frequency-locked loops) and many other “pendulum-like” systems. Similar to the usual pendulum equation, such systems are typically featured by infinite sequences of equilibria points, and none of which can be globally asymptotically stable. A natural extension of the global asymptotic stability is the gradient-like behavior, that is, convergence of any solution to one of the equilibria. In this paper, we offer an efficient frequency-domain criterion for gradientlike behavior. This criterion is not only applicable to a broad class of infinite-dimensional systems with periodic nonlinearities, but in fact ensures the equilibria set stability under singular perturbation. In particular, the proposed criterion guarantees the absence of periodic solutions that are considered to be undesirable in synchronization systems. In this paper we also discuss a relaxed version of this criterion, which guarantees the absence of “high-frequency” periodic solutions, whose frequencies lie beyond a certain bounded interval.