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Bursting dynamics and the zero-Hopf bifurcation of simple jerk system

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  • Sun, Xi
  • Yan, Shaohui
  • Zhang, Yuyan
  • Wang, Ertong
  • Wang, Qiyu
  • Gu, Binxian

Abstract

We characterize the zero-Hopf bifurcation at a singular point of a parameter jerk system. By employing the second order averaging theory, we demonstrate that up to three periodic orbits generated as disturbance parameters tend to zero. Again, both the bifurcation mechanism and bursting dynamics of the 3D jerk system with external periodic excitation are systematically explored. While an order difference exists between the frequency of external excitation and the average frequency of the system, the system exhibits bursting oscillations. The mechanisms of different bursting oscillations are investigated by means of the equilibrium point curve and the transformed phase portraits. As the amplitudes of the excitation change, the system displays “delayed symmetric pitchfork/point, delayed symmetric pitchfork/supHopf, and delayed pitchfork/supHopf/homoclinic connection bursting”. Finally, an analog circuit is designed to verify the complex bursting phenomena of the system.

Suggested Citation

  • Sun, Xi & Yan, Shaohui & Zhang, Yuyan & Wang, Ertong & Wang, Qiyu & Gu, Binxian, 2022. "Bursting dynamics and the zero-Hopf bifurcation of simple jerk system," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    • Handle: RePEc:eee:chsofr:v:162:y:2022:i:c:s0960077922006658
    DOI: 10.1016/j.chaos.2022.112455
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    References listed on IDEAS

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    1. Njitacke, Z.T. & kengne, J. & Kengne, L. Kamdjeu, 2017. "Antimonotonicity, chaos and multiple coexisting attractors in a simple hybrid diode-based jerk circuit," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 77-91.
    2. 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.
    3. Zhang, Shaohua & Zhang, Hongli & Wang, Cong & Ma, Ping, 2020. "Bursting oscillations and bifurcation mechanism in a permanent magnet synchronous motor system with external load perturbation," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    4. Kengne, Jacques & Mogue, Ruth Line Tagne & Fozin, Theophile Fonzin & Telem, Adelaide Nicole Kengnou, 2019. "Effects of symmetric and asymmetric nonlinearity on the dynamics of a novel chaotic jerk circuit: Coexisting multiple attractors, period doubling reversals, crisis, and offset boosting," Chaos, Solitons & Fractals, Elsevier, vol. 121(C), pages 63-84.
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