We present results on the bifurcations that occur in a two-parameter family of ordinary differential equations in R4 that describe a pair of linearly coupled oscillators. In particular, we describe the existence of infinitely many stability zones in parameter space in which two stable periodic orbits coexist.
Bibliographical noteFunding Information:
While much is known about the bifurcations that occur in the solution set of a forced oscillator as the frequency and amplitude of the forcing are varied, the behavior of the solutions of coupled oscillators as the natural frequency and the coupling strength are varied has not been completely characterized in any system. In general some results can be obtained at very weak or very strong coupling by asymptotic methods, but numerical studies \[1-3\] show that most of the interesting effects of coupling occur at intermediate coupling strengths. In this paper we give some results for a model system of two coupled oscillators in which many results at moderate coupling strengths and frequencies can be obtained analytically. In particular, we show how an infinite sequence of resonance horns and bifurcations from a periodic orbit arises, and give results that suggest that the Arnol'd structure for two- 1 Supported in part by NIH Grant 29123 and a grant from the University of Utah.
2 Supported in part by NSF Grant DMS 83-01247. Supported in part by NSERC (Canada) A4274 and FCAC (Quebec) EQ1438.