The saddle case of Rayleigh–Duffing oscillators

Abstract

The global dynamics of Rayleigh–Duffing oscillators $$\ddot{x}+a\dot{x}+b\dot{x}^3+cx+dx^3=0$$ , where $$(a,b,c,d)\in \{(a,b,c,d)\in \mathbb {R}^4: b\ne 0,~d>0\}$$ , have been investigated in Chen and Zou (J Phys A 49:165202, 2016). In this paper, the complement case $$(a,b,c,d)\in \{(a,b,c,d)\in \mathbb {R}^4: b\ne 0,~d<0\}$$ will be completely studied, where the bifurcation diagram includes pitchfork bifurcation, Hopf bifurcation, and heteroclinic bifurcation. Meanwhile, the global phase portraits in the Poincare disc are given. The system has at most one limit cycle. Moreover, when the limit cycle exists, its corresponding parameter region lies between Hopf and heteroclinic loop bifurcation curves in the parametric space. In addition, the analytic properties of the heteroclinic loop bifurcation curve are also analyzed. Finally, a few numerical examples are presented to verify our theoretical results.