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On Hopf and Fold Bifurcations of Jerk Systems

Author

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  • Cristian Lăzureanu

    (Department of Mathematics, Politehnica University Timişoara, P-ta Victoriei 2, 300006 Timişoara, Romania
    Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timişoara, Parvan Blv. 4, 300223 Timişoara, Romania
    These authors contributed equally to this work.)

  • Jinyoung Cho

    (Department of Mathematics, Politehnica University Timişoara, P-ta Victoriei 2, 300006 Timişoara, Romania
    Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timişoara, Parvan Blv. 4, 300223 Timişoara, Romania
    These authors contributed equally to this work.)

Abstract

In this paper we consider a jerk system x ˙ = y , y ˙ = z , z ˙ = j ( x , y , z , α ) , where j is an arbitrary smooth function and α is a real parameter. Using the derivatives of j at an equilibrium point, we discuss the stability of that point, and we point out some local codim-1 bifurcations. Moreover, we deduce jerk approximate normal forms for the most common fold bifurcations.

Suggested Citation

  • Cristian Lăzureanu & Jinyoung Cho, 2023. "On Hopf and Fold Bifurcations of Jerk Systems," Mathematics, MDPI, vol. 11(20), pages 1-15, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4295-:d:1260105
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    References listed on IDEAS

    as
    1. Zhang, Sen & Zeng, Yicheng, 2019. "A simple Jerk-like system without equilibrium: Asymmetric coexisting hidden attractors, bursting oscillation and double full Feigenbaum remerging trees," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 25-40.
    2. 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.
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    Cited by:

    1. Cristian Lăzureanu & Jinyoung Cho, 2024. "On a Family of Hamilton–Poisson Jerk Systems," Mathematics, MDPI, vol. 12(8), pages 1-12, April.
    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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