The focus case of a nonsmooth Rayleigh–Duffing oscillator

Zhaoxia Wang,Yilei Tang,Hebai Chen

doi:10.1007/s11071-021-07007-9

Abstract

In this paper, we study the global dynamics of a nonsmooth Rayleigh–Duffing equation $$\ddot{x}+a{\dot{x}}+b{\dot{x}}|{\dot{x}}|+cx+{\mathrm {d}}x^3=0$$ 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.

Highlights

Introduction and main resultsIn the middle of 17th century, C
We compare the global dynamics for the focus case of nonsmooth Rayleigh-Duffing oscillator (1.2b) with smooth Rayleigh-Duffing oscillator, smooth van der Pol-Duffing oscillator and nonsmooth van der Pol-Duffing oscillator
In the case of having three equilibria, a homoclinic bifurcation, a double small limit cycle bifurcation and a double large limit cycle bifurcation will occur in both systems (1.2b) and (6.1)

Summary

Introduction and main results

There is a unique limit cycle occurring in a small neighborhood of El Notice that E0 is a stable nilpotent focus when μ1 = μ2 = 0, where limit cycles may be bifurcated. The unique stable limit cycle still persists when μ2 = 0 This is a contradiction, which completes the proof. When μ1 > 0, by Proposition 2.1 the stable weak foci El and Er become unstable rough foci and two limit cycles occur at the same time, one is in a small neighborhood of El and the other is in a small neighborhood of Er, as μ2 changes from 0 to a negative value. Stable nilpotent focus) and one stable limit cycle occurs in a small neighborhood of E0 as μ2 changes to a negative value from 0.

Limit cycles and homoclinic loops

Numerical examples and discussions

Nonsmooth and smooth Rayleigh-Duffing oscillator

Nonsmooth Rayleigh-Duffing oscillator and van der Pol-Duffing oscillator