Frequency Plateaus in a Chain of Weakly Coupled Oscillators, I.

A chain of $n + 1$ weakly coupled oscillators with a linear gradient in natural frequencies is shown to exhibit “frequency plateaus,” or sequences of oscillators having the same frequency, with a jump in frequency from one plateau to another. We first show that the equations for the coupled oscillators admit an invariant $(n + 1)$-torus on which the equations have a special form, one in which an n-dimensional subsystem is approximately invariant. We then show that when the linear gradient becomes too steep to allow phaselocking, there emerges a large-scale invariant circle in this n-dimensional system which corresponds to the existence of a pair of plateaus, and whose homotopy class within the n-torus corresponds to the position of the frequency jump. Also discussed are the effects of anisotropic and nonuniform coupling.