Convergent and oscillatory solutions in infinite-dimensional synchronization systems

Control systems that arise in phase synchronization problems are featured by infinite sets of stable and unstable equilibria, caused by presence of periodic nonlinearities. For this reason, such systems are often called “pendulum-like”. Their dynamics are thus featured by multi-stability and cannot be examined by classical methods that have been developed to test the lobal stability of a unique equilibrium point. In general, only sufficient conditions for the solution convergence are known that are usually derived for pendulum-like systems of Lurie type, that is, interconnections of stable LTI blocks and periodic nonlinearities, which obey sector or slope restrictions. Most typically, these conditions are written as multi-parametric frequency-domain inequalities, which should be satisfied by the transfer function of the system’s linear part. Remarkably, if the frequencydomain inequalities hold outside some bounded range of frequencies, then the absence of periodic solutions with frequencies in this range is guaranteed, which can be considered as a weaker asymptotical property. It should be noticed that validation of the frequency domain stability condition for a given structure of the known linear part of the system is a self-standing nontrivial problem. In this paper, we demonstrate that a previously derived frequency-domain conditions for stability and absence of oscillations can be substantially simplified, parameters ensuring the corresponding asymptotic property. We demonstrate the efficiency of new criteria on specific synchronization systems.