The focus case of a nonsmooth Rayleigh–Duffing oscillator

In this paper, we study the global dynamics of a nonsmooth Rayleigh–Duffing equation x ¨ + a x ˙ + b x ˙ | x ˙ | + c x + d x 3 = 0 for the case d > 0 , i.e., the focus case. The global dynamics of this nonsmooth Rayleigh–Duffing oscillator for the case d < 0 , i.e., the saddle case, has been studied in the companion volume (Wang and Chen in Int J Non-Linear Mech 129: 103657, 2021). The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results.