Stability and oscillations of singularly perturbed phase synchronization systems with distributed parameters

In this paper we examine singularly perturbed systems, described by integro-differential Volterra equations with periodic nonlinearities and a small parameter at the higher derivative. This type of equations describes many “pendulumlike” systems as well as phase-locked loops and other synchronization circuits. Such systems usually have infinite sequence of equilibrium points. The main problem for this class of control systems is the problem of gradient-like behavior, i.e. whether any solution converges to one of the equilibria. In this paper we propose a frequency-algebraic criterion of gradient-like behavior for singular perturbed systems. If the system is not gradient-like it may have periodic regimes. 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.