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Homotopy homoclinic orbit and global dynamics analysis of the double-excited Duffing–van der Pol system

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Listed:
  • Zhang, Danjin
  • Ren, Meirong
  • Qian, Youhua

Abstract

The chaos is a complex phenomenon that appears random but actually possesses inherent patterns. The study of chaos is significant for understanding many complex phenomena in nature. This paper aims to explore the threshold curves of the chaos phenomenon by analyzing the homoclinic orbits and global dynamics of the doubly-excited Duffing–van der Pol system using the homotopy analysis method and Melnikov function analysis. Through the homotopy analysis method, the third-order approximate homotopy homoclinic solution of the system’s precise homoclinic solution is provided in a specific form. By substituting the third-order approximate homotopy homoclinic solution into the Melnikov function of homoclinic bifurcation, the chaotic phenomena when the two excitation frequencies are equal and unequal are analyzed. Through the obtained chaotic threshold curves, we present the phenomena of periodic and chaotic motions separately and draw phase diagrams, Poincaré maps, Lyapunov exponent diagrams, and spectrum diagrams for verification.

Suggested Citation

  • Zhang, Danjin & Ren, Meirong & Qian, Youhua, 2025. "Homotopy homoclinic orbit and global dynamics analysis of the double-excited Duffing–van der Pol system," Chaos, Solitons & Fractals, Elsevier, vol. 199(P1).
    • Handle: RePEc:eee:chsofr:v:199:y:2025:i:p1:s0960077925005934
    DOI: 10.1016/j.chaos.2025.116580
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    References listed on IDEAS

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    1. Guo, Xiuying & Tian, Ruilan & Xue, Qiang & Zhang, Xiaolong, 2022. "Sub-harmonic Melnikov function for a high-dimensional non-smooth coupled system," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    2. Akyildiz, F. Talay & Vajravelu, K. & Liao, S.-J., 2009. "A new method for homoclinic solutions of ordinary differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1073-1082.
    3. Wang, Meiqi & Zhao, Jingyan & Wang, Ruichen & Qin, Chengwei & Liu, Pengfei, 2025. "Dynamical characterisation of fractional-order Duffing-Holmes systems containing nonlinear damping under constant simple harmonic excitation," Chaos, Solitons & Fractals, Elsevier, vol. 190(C).
    4. Y. J. F. Kpomahou & A. Adomou & J. A. Adéchinan & A. E. Yamadjako & I. V. Madogni & Jesus M. Munoz-Pacheco, 2022. "Chaotic Behaviors and Coexisting Attractors in a New Nonlinear Dissipative Parametric Chemical Oscillator," Complexity, Hindawi, vol. 2022, pages 1-16, March.
    5. Shijun Liao, 2012. "Homotopy Analysis Method in Nonlinear Differential Equations," Springer Books, Springer, number 978-3-642-25132-0, January.
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