BIFURCATION ANALYSIS OF THE QUASI-PERIODIC SOLUTION WITH THREE-PHASE SYNCHRONIZED ENVELOPES IN A RING OF THREE-COUPLED, BISTABLE OSCILLATORS

Abstract

In this study, we investigate the bifurcation of the quasi-periodic solution in a ring of three-coupled, bistable oscillators. The quasi-periodic solution under study is a propagating wave solution with three-phase synchronized envelope, and we name it "ICC3ϕ" which means invariant closed curve for which time waveforms have three-phase synchronized envelope. We obtain the two-parameter bifurcation diagram of ICC3ϕ with respect to the coupling factor α versus the nonlinear strength ϵ. Consequently, we detect several Arnold tongues showing various periodic solutions of ICC3ϕ. By investigating the bifurcation on the boundary of these Arnold tongues, we clarify the transition from periodic to quasi-periodic solution of ICC3ϕ or the inverse. Specifically, the transition of the period-4, period-5, and period-7 solutions to ICC3ϕ is due to a saddle-node bifurcation, which has no hysteresis. In contrast, the transition from the period-3 solution to ICC3ϕ is a combination of subcritical pitchfork and heteroclinic bifurcations; hence, it has hysteresis. We draw diagrams showing the connection of unstable manifolds of heteroclinic bifurcation.