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Multi-Frequency Homotopy Analysis Method for Coupled Van der Pol-Duffing System with Time Delay

Author

Listed:
  • Youhua Qian

    (College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China)

  • Shuli Wang

    (College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China)

  • Shuping Chen

    (College of Mathematics, Xiamen University of Technology, Xiamen 361024, China)

Abstract

This paper mainly studied the analytical solutions of three types of Van der Pol-Duffing equations. For a system with parametric excitation frequency, we knew that the ordinary homotopy analysis method would be unable to find the analytical solution. Thus, we primarily used the multi-frequency homotopy analysis method (MFHAM). First, the MFHAM was introduced, and the solution of the system was expressed by constructing auxiliary linear operators. Then, the method was applied to three specific systems. We compared the numerical solution obtained using the Runge–Kutta method with the analytical solution to verify the correctness of the latter. Periodic solutions, with and without time delay, were also compared under the same parameters. The results demonstrated that it was both effective and correct to use the MFHAM to find analytical solutions to Van der Pol-Duffing systems, which were classical systems. By comparison, the MFHAM proved to be effective for time delay systems.

Suggested Citation

  • Youhua Qian & Shuli Wang & Shuping Chen, 2023. "Multi-Frequency Homotopy Analysis Method for Coupled Van der Pol-Duffing System with Time Delay," Mathematics, MDPI, vol. 11(2), pages 1-17, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:2:p:407-:d:1033888
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    References listed on IDEAS

    as
    1. Hu, H.Y. & Wang, Z.H., 2009. "Singular perturbation methods for nonlinear dynamic systems with time delays," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 13-27.
    2. Kimiaeifar, A. & Saidi, A.R. & Bagheri, G.H. & Rahimpour, M. & Domairry, D.G., 2009. "Analytical solution for Van der Pol–Duffing oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2660-2666.
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