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Numerical investigation of the origin of compound relaxation oscillations in a nonlinear oscillator

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Abstract

The phenomenon of relaxation oscillations is a typical fast–slow dynamical behavior. In this paper, we take a type of nonlinear oscillator involving multiple coexisting attractors as an example and aim to reveal interesting patterns of relaxation oscillations, namely the so-called compound relaxation oscillations. To begin with, a relaxation oscillation pattern with asymmetrical transitions is obtained. Then, a two-parameter bifurcation diagram is plotted to explore the transitions of relaxation oscillations. It is found that the number and stability of the attractors near fold bifurcation points may change when the system parameters vary. This is manifested in two ways: First, when a fold bifurcation occurs in the upper equilibrium branch, the system is bi-stable; while a fold bifurcation occurs in the lower equilibrium branch, the system is stable. Second, the system is bi-stable when fold bifurcations occur in the upper and lower equilibrium branches. For the above two cases, we use fast–slow analysis and attraction domain analysis to explore the dynamical mechanisms of the relaxation oscillation behaviors. As a result, several different relaxation oscillation patterns are obtained. In particular, relaxation oscillations with asymmetric structure, i.e., compound relaxation oscillations, characterized by the system trajectory transitions passing through the middle branch at one end and crossing it at the other, are discovered and researched.

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Data availability

The data used to support the finding of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors express their gratitude to the anonymous reviewers whose comments and suggestions have helped improve this paper. This work is supported by the National Natural Science Foundation of China (G. Nos. 12072132 and 12272150) and the Qing Lan Project of Jiangsu Province.

Funding

Funding for this study was obtained from the National Natural Science Foundation of China, 12072132 and 12272150, Xiujing Han, Qinglan Project of Jiangsu Province of China.

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Authors and Affiliations

  1. Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang, 212013, People’s Republic of China

    Muchuan Ding, Xiujing Han & Qinsheng Bi

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  1. Muchuan Ding
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  2. Xiujing Han
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  3. Qinsheng Bi
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Contributions

M.D. was involved in conceptualization, formal analysis, investigation, methodology, and writing—original draft. X.H. was responsible for conceptualization, supervision, project administration, and writing—reviewing and editing. Q.B. contributed to methodology, supervision, and validation.

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Correspondence to Xiujing Han.

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Ding, M., Han, X. & Bi, Q. Numerical investigation of the origin of compound relaxation oscillations in a nonlinear oscillator. Nonlinear Dyn 111, 13853–13864 (2023). https://doi.org/10.1007/s11071-023-08576-7

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