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Local stability and Hopf bifurcations analysis of the Muthuswamy-Chua-Ginoux system

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A Correction to this article was published on 06 May 2022

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Abstract

The three-dimensional Muthuswamy–Chua–Ginoux (MCG, for short) circuit system based on a thermistor is a generalization of the classical Muthuswamy–Chua circuit differential system. At present, there are only partial numerical simulations for the qualitative analysis of the MCG circuit system. In this work, we study local stability and Hopf bifurcations of the MCG circuit system depending on 8 parameters. The emerging of limit cycles under zero-Hopf bifurcation and Hopf bifurcation is investigated in detail by using the averaging method and the center manifolds theory, respectively. We provide sufficient conditions for a class of the circuit systems to have a prescribed number of limit cycles bifurcating from the zero-Hopf equilibria by making use of the third-order averaging method, as well as the methods of Gröbner basis and real solution classification from symbolic computation. Such algebraic analysis allows one to study the zero-Hopf bifurcation for any other differential system in dimension 3 or higher. After, the classical Hopf bifurcation of the circuit system is analyzed by computing the first three focus quantities near the Hopf equilibria. Some examples and numerical simulations are presented to verify the established theoretical results.

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Acknowledgements

We are very grateful to the anonymous referees whose constructive comments and suggestions helped improve and clarify this paper. This research is supported by the National Natural Science Foundation of China (No.12101032, No.12131004, No.11790273 and No.11801582) and Guangdong Basic and Applied Basic Research Foundation (No.2019A1515011239).

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  1. Yuzhou Tian and Bo Huang contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Yuzhou Tian

  2. LMIB - School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    Bo Huang

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  1. Yuzhou Tian
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Correspondence to Bo Huang.

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Tian, Y., Huang, B. Local stability and Hopf bifurcations analysis of the Muthuswamy-Chua-Ginoux system. Nonlinear Dyn 109, 1135–1151 (2022). https://doi.org/10.1007/s11071-022-07409-3

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