Phase Transitions and Other Phenomena in Chains of Coupled Oscillators

Abstract

A very broad framework is given for the investigation of long chains of N weakly coupled oscillators. The framework allows nonmonotonic changes of natural frequency along the chain, differences in coupling strengths, anisotropy in the two directions of coupling, and very general local coupling functions. It is shown that the phase locked solutions of all the systems of oscillators within this framework converge for large N to solutions of a class of nonlinear, singularly perturbed, two-point boundary value problems. Using the latter continuum equations, it is also shown that there are parameter regimes in which the solutions have qualitatively different behavior, with a phase-transition-like change in behavior across the boundary between parameter regimes in the limit $N \to \infty $. Special important cases are discussed, including the effects of local changes in frequencies, local changes in coupling strengths, and different kinds of anisotropy.