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Global study of Rayleigh–Duffing oscillators

In this paper we investigate the global dynamics of Rayleigh–Duffing oscillators with global parameters, including equilibria at both finity and infinity, existences and coexistence of limit cycles and homoclinic loops. In fact, this oscillator will occur Hopf bifurcations, homoclinic bifurcations and double limit cycle bifurcations. Moreover, we find that the homoclinic bifurcation of this oscillator is special which is a gluing bifurcation. The global bifurcation diagram and all phase portrait are given, and numerical simulations are shown to verify our analysis finally.