In this paper, systems of nonlinear integro-differential Volterra equations are examined that can be represented as feedback interconnections of linear time-invariant block and periodic nonlinearities. The interest in such systems is motivated by their numerous applications in mechanical, electrical and communication engineering; examples include, but are not limited to, models of phase-locked loops, pendulum-like mechanical systems, coupled vibrational units and electric machines. Systems with periodic nonlinearities are usually featured by multistability and have infinite sequences of (locally) stable and unstable equilibria; their trajectories may exhibit nontrivial (e.g. chaotic) behavior. We offer frequency-domain criteria, ensuring convergence of any solution to one of the equilibria points, and this property is referred to as the gradient-like behavior and corresponds to phase locking in synchronization systems. Although it is hard to find explicitly the equilibrium, attracting a given trajectory, we give a constructive estimate for the distance between this limit equilibrium and the initial condition. The relevant estimates are closely related to the analysis of cycle slipping in synchronization systems. In the case where the criterion of gradient-type behavior fails, a natural question arises-which nonconverging solutions may exist in the system and, in particular, how many periodic solutions it has. We show that a relaxation of the frequency-domain convergence criterion ensures the absence of high-frequency periodic orbits. The results obtained in the paper are based on the method of integral quadratic constraints that has arisen in absolute stability theory and stems from Popov's techniques of "a priori integral indices". We illustrate the analytic results by numerical simulations.

Systems that can be decomposed as feedback interconnections of stable linear blocks and periodic nonlinearities arise in many physical and engineering applications. The relevant models e.g. describe oscillations of a viscously damped pendulum, synchronization circuits (phase, frequency and delay locked loops) and networks of coupled power generators. A system with periodic nonlinearities usually has multiple equilibria (some of them being locally unstable). Many tools of classical stability and control theories fail to cope with such systems. One of the efficient methods, elaborated to deal with periodic nonlinearities, stems from the celebrated Popov method of 'integral indices', or integral quadratic constraints; this method leads, in particular, to frequency-domain criteria of the solutions' convergence, or, equivalently, global stability of the equilibria set. In this paper, we further develop Popov's method, addressing the problem of robustness of the convergence property against external disturbances that do not oscillate at infinity (allowing the system to have equilibria points). Will the forced solutions also converge to one of the equilibria points of the disturbed system? In this paper, a criterion for this type of robustness is offered.

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.