Global behaviors of pendulum-like dynamical networks

This paper focuses on the global behaviors of a class of pendulum-like dynamical networks with multiple equilibria. Sufficient conditions for the determination of gradient-like as well as dichotomy behavior are given in terms of linear matrix inequalities (LMIs). In order to examine the global synchronous behavior, a concept of generalized synchronization is introduced, and the criterion for generalized phase synchronization is obtained in a simple form. The derived results are of particular convenience for networks that possess large numbers of nodes, and they can be used to discuss controller design problems as well. Applications to interconnected phase-locked loops (PLLs) are given at the end of the paper to demonstrate the validity of the proposed approach. It is shown that by investigating the global behaviors of the phase variable, our results suggest a viable and effective way for the phase-locking phenomenon.