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Bursting oscillations in coupling Mathieu-van der Pol oscillator under parametric excitation

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  • Jiang, Y.D.
  • Zhang, W.
  • Zhang, Y.F.
  • Bi, Q.S.

Abstract

The bursting oscillation is a fast-slow oscillation. A complex four-dimensional coupled Mathieu-van der Pol oscillator is analyzed to investigate the bursting oscillations. The multiscale phenomena appear in four-dimensional coupled Mathieu-van der Pol oscillator due to the significant difference between the natural frequencies of the system and frequencies of the parametric excitation. Analyzing the bifurcation diagrams of four-dimensional coupled Mathieu-van der Pol oscillator using the bifurcation theory and fast-slow analysis, we identify four different bursting oscillation modes. Additionally, an intriguing phenomenon is observed as the parameters of the fast and slow systems change in the orbits. This phenomenon is called as the slow channel effect. The trajectory traverses the bifurcation point without the immediate bifurcation behaviors but gradually converges to a stable limit cycle after an apparent delay. Four-dimensional coupled Mathieu-van der Pol oscillator has a composite Hopf bifurcation which means that a Hopf bifurcation respectively corresponds to the large and small limit cycles. This research provides the insights into the mechanism of the bursting oscillations.

Suggested Citation

  • Jiang, Y.D. & Zhang, W. & Zhang, Y.F. & Bi, Q.S., 2024. "Bursting oscillations in coupling Mathieu-van der Pol oscillator under parametric excitation," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    • Handle: RePEc:eee:chsofr:v:178:y:2024:i:c:s0960077923011815
    DOI: 10.1016/j.chaos.2023.114279
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    1. Wang, Shaojie & He, Shaobo & Yousefpour, Amin & Jahanshahi, Hadi & Repnik, Robert & Perc, Matjaž, 2020. "Chaos and complexity in a fractional-order financial system with time delays," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    2. Hassan Kamil Jassim & Mohammed Abdulshareef Hussein, 2023. "A New Approach for Solving Nonlinear Fractional Ordinary Differential Equations," Mathematics, MDPI, vol. 11(7), pages 1-13, March.
    3. Danjin Zhang & Youhua Qian, 2023. "Bursting Oscillations in General Coupled Systems: A Review," Mathematics, MDPI, vol. 11(7), pages 1-16, April.
    4. Semmler, Willi, 1995. "Solving Nonlinear Dynamic Models by Iterative Dynamic Programming," Computational Economics, Springer;Society for Computational Economics, vol. 8(2), pages 127-154, May.
    5. Guanrong Chen & Jin-quing Fang & Yiguang Hong & Huashu Qin, 2000. "Controlling hopf bifurcations: Discrete-time systems," Discrete Dynamics in Nature and Society, Hindawi, vol. 5, pages 1-5, January.
    6. Duan, Daifeng & Niu, Ben & Wei, Junjie, 2019. "Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 206-216.
    7. Zhang, Shaohua & Wang, Cong & Zhang, Hongli & Ma, Ping & Li, Xinkai, 2022. "Dynamic analysis and bursting oscillation control of fractional-order permanent magnet synchronous motor system," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    8. Giresse, Tene Alain & Crépin, Kofane Timoleon, 2017. "Chaos generalized synchronization of coupled Mathieu-Van der Pol and coupled Duffing-Van der Pol systems using fractional order-derivative," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 88-100.
    9. Chen, Z. & Yu, P., 2005. "Controlling and anti-controlling Hopf bifurcations in discrete maps using polynomial functions," Chaos, Solitons & Fractals, Elsevier, vol. 26(4), pages 1231-1248.
    10. Wang, Lingyu & Sun, Kehui & Peng, Yuexi & He, Shaobo, 2020. "Chaos and complexity in a fractional-order higher-dimensional multicavity chaotic map," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    11. Qi-Ming Chen & Michael Fischer & Yuki Nojiri & Michael Renger & Edwar Xie & Matti Partanen & Stefan Pogorzalek & Kirill G. Fedorov & Achim Marx & Frank Deppe & Rudolf Gross, 2023. "Quantum behavior of the Duffing oscillator at the dissipative phase transition," Nature Communications, Nature, vol. 14(1), pages 1-7, December.
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