Summary
We consider asymptotic behavior of integro-differential equations with periodic nonlinearities, featured by infinite sequences of equilibria. Such systems are also referred to as ``synchronization systems'' in view of numerous applications to dynamics of control circuits such as phase-locked, frequency-locked, and delay-locked loops. In these applications, one is typically interested in gradient-like behavior of a system, that is, the convergence of all solutions to equilibria and the absence of oscillations. In this talk, new frequency-domain conditions are obtained to ensure gradient-like behavior. Remarkably, the relaxation of these conditions to the subinterval of frequencies guarantees the absence of undesirable high-frequency oscillations in the system.
