Dynamics in a Chain of Overdamped Pendula Driven by Constant Torques

Abstract

This paper studies the dynamical behavior of a chain of overdamped pendula driven by constant torques with nearest neighbor coupling. The coupling constant K is assumed to be $>0$, independent of N. It is shown that when the system does not have equilibrium points, the global attractor of this system is a one-dimensional closed curve, so no matter what input frequencies $\omega_j$ are used, the existence, uniqueness, and global stability of a limit cycle of second kind are proved; therefore, any solution will be frequency locked in the long time limit. On the other hand, if there are equilibrium points in the system, any solution is bounded and converges to an equilibrium point.