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Bursting Dynamics in a Singular Vector Field with Codimension Three Triple Zero Bifurcation

Author

Listed:
  • Weipeng Lyu

    (Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China
    Faculty of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China)

  • Shaolong Li

    (Faculty of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China)

  • Zhenyang Chen

    (Faculty of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China)

  • Qinsheng Bi

    (Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China)

Abstract

As a kind of dynamical system with a particular nonlinear structure, a multi-time scale nonlinear system is one of the essential directions of the current development of nonlinear dynamics theory. Multi-time scale nonlinear systems in practical applications are often complex forms of coupling of high-dimensional and high codimension characteristics, leading to various complex bursting oscillation behaviors and bifurcation characteristics in the system. For exploring the complex bursting dynamics caused by high codimension bifurcation, this paper considers the normal form of the vector field with triple zero bifurcation. Two kinds of codimension-2 bifurcation that may lead to complex bursting oscillations are discussed in the two-parameter plane. Based on the fast–slow analysis method, by introducing the slow variable W = A sin ( ω t ) , the evolution process of the motion trajectory of the system changing with W was investigated, and the dynamical mechanism of several types of bursting oscillations was revealed. Finally, by varying the frequency of the slow variable, a class of chaotic bursting phenomena caused by the period-doubling cascade is deduced. Developing related work has played a positive role in deeply understanding the nature of various complex bursting phenomena and strengthening the application of basic disciplines such as mechanics and mathematics in engineering practice.

Suggested Citation

  • Weipeng Lyu & Shaolong Li & Zhenyang Chen & Qinsheng Bi, 2023. "Bursting Dynamics in a Singular Vector Field with Codimension Three Triple Zero Bifurcation," Mathematics, MDPI, vol. 11(11), pages 1-20, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:11:p:2486-:d:1158069
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    References listed on IDEAS

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    1. Xue, Miao & Gou, Junting & Xia, Yibo & Bi, Qinsheng, 2021. "Computation of the normal form as well as the unfolding of the vector field with zero-zero-Hopf bifurcation at the origin," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 377-397.
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    Cited by:

    1. Cristian Lăzureanu & Jinyoung Cho, 2023. "On Hopf and Fold Bifurcations of Jerk Systems," Mathematics, MDPI, vol. 11(20), pages 1-15, October.
    2. Cristian Lăzureanu, 2023. "On the Double-Zero Bifurcation of Jerk Systems," Mathematics, MDPI, vol. 11(21), pages 1-12, October.

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