Nonlinear Dynamics

, Volume 93, Issue 4, pp 2283–2300 | Cite as

The saddle case of Rayleigh–Duffing oscillators

  • Hebai Chen
  • Deqing HuangEmail author
  • Yupei Jian
Original Paper


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 Poincaré 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.


Rayleigh–Duffing oscillator Limit cycle 2-Saddle loop Bifurcation diagram 

Mathematics Subject Classification

34C07 34C23 34C37 34K18 



Research is partially supported by National Natural Science Foundation of China under Grant Nos. 11572263, 61773323, 61603316, 61433011. The authors are grateful to the reviewers for their helpful suggestions and comments.


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Authors and Affiliations

  1. 1.College of Mathematics and Computer ScienceFuzhou UniversityFuzhouPeople’s Republic of China
  2. 2.Key Laboratory of Operations Research and Control of University in FujianFuzhou UniversityFuzhouPeople’s Republic of China
  3. 3.Institute of Systems Science and Technology, School of Electrical EngineeringSouthwest Jiaotong UniversityChengduPeople’s Republic of China

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