Dynamic Behavior of a Large Ring of Coupled Self-Excited Oscillators

Abstract

Abstract Interconnected, self-excited oscillators are often found in nature and in engineered devices. In this work, a ring of van der Pol oscillators, each of which is connected to its immediate neighbors, is considered. The focus is on the emergent behavior of a large number of oscillators. Conditions are determined under which time-independent solutions are obtained, and the linear stability of these solutions is investigated. The effect of the singularity of the coupling matrix on the ring dynamics is explored. When this becomes singular, an infinite number of steady states is present, and the phenomenon of oscillation death arises. It is also possible to have, depending on initial conditions, all oscillators with in-phase synchrony, metachronal traveling waves with different wavelengths going around the ring, or standing waves. Interconnected oscillators can propagate information at a group velocity, and the information signal is present as an amplitude modulation.