Dynamics of a large number of coupled, self-excited oscillators in a ring

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. The conditions under which time-independent solutions are obtained are determined, 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 arises and the phenomenon of oscillation death is possible. Numerically-calculated dynamics of the coupled oscillators for small and large damping show complicated global behavior. Of interest are the longtime and large number of oscillators limit. It is possible to have, depending on initial conditions, all oscillators with in-phase synchrony, or metachronal traveling waves with different wavelengths going around the ring. Linear combinations, such as standing waves, are also observed. Interconnected oscillators can propagate information at a group velocity; the individual oscillators vibrate at a carrier frequency, and the signal as an amplitude modulation of that.