We study a two-parameter family of ordinary differential equations in R4 that governs the dynamics of two coupled planar oscillators. Each oscillator has a unique periodic solution that is attracting and the uncoupled product system has a unique invariant torus that is attracting. The torus persists for weak coupling and contains two periodic solutions when the coupling is linear and conservative. One of these, in which the oscillators are synchronized, persists and is stable for all coupling strengths. The other, in which the oscillators are π radiant out of phase, disappears either in a Hopf bifurcation or when fixed points appear on the orbit at a critical ratio of the coupling strength to the frequency. The out-of-phase oscillation is unstable except on an open set in the frequency-coupling-strength plane which contains moderate values of both parameters. Furthermore, there are tori bifurcating from the out-of-phase solution, which means, according to the Arnol'd theory for Hopf bifurcations in maps, that there may be periodic solutions of arbitrarily large period and chaotic solutions as well. Numerous other bifurcations occur, and there are a number of higher codimension singularities. In a large region of the frequency-coupling parameter plane stable steady states coexist with stable periodic solutions.
Bibliographical noteFunding Information:
4.7. Isoclines 4.8. Rotation number and Floquet multipliers near infinite period bifurcation with fl ~ (1/2,1) 4.9. Variational flow near infinite period bifurcation with B ~ (0,1/2) 4.10. Bifurcation curves 4.11. Bifurcations from steady states in H 5. Numerical results 5.1. Steady states 5.2. Periodic solutions and their bifurcations for 8 < fl/2 5.3. Periodic solutions and their bifurcations for 8 > fl/2 6. Discussion References Appendix 1 Reduction of the steady state equations Appendix 2 Numerical software *Supported in part by NSF Grant #DMS-83-01247. tSupported in part by NSERC (Canada) ~A4274 and FCAC (Quebec) #EQ1438. *Supported in part by NIH Grant ~29123 and a grant from the University of Utah.
Copyright 2014 Elsevier B.V., All rights reserved.