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Semi-Analytical Closed-Form Solutions for the Rikitake-Type System through the Optimal Homotopy Perturbation Method

Author

Listed:
  • Remus-Daniel Ene

    (Department of Mathematics, Politehnica University of Timisoara, 300006 Timisoara, Romania
    These authors contributed equally to this work.)

  • Nicolina Pop

    (Department of Physical Foundations of Engineering, Politehnica University of Timisoara, 300223 Timisoara, Romania
    These authors contributed equally to this work.)

Abstract

The goal of this work is to build semi-analytical solutions of the Rikitake-type system by means of the optimal homotopy perturbation method (OHPM) using only two iterations. The chaotic behaviors are excepted. By taking into consideration the geometrical properties of the Rikitake-type system, the closed-form solutions can be established. The obtained solutions have a periodical behavior. These geometrical properties allow reducing the initial system to a second-order nonlinear differential equation. The latter equation is solved analytically using the OHPM procedure. The validation of the OHPM method is presented for three cases of the physical parameters. The advantages of the OHPM technique, such as the small number of iterations (the efficiency), the convergence control (in the sense that the semi-analytical solutions are approaching the exact solution), and the writing of the solutions in an effective form, are shown graphically and with tables. The accuracy of the results provides good agreement between the analytical and corresponding numerical results. Other dynamic systems with similar geometrical properties could be successfully solved using the same procedure.

Suggested Citation

  • Remus-Daniel Ene & Nicolina Pop, 2023. "Semi-Analytical Closed-Form Solutions for the Rikitake-Type System through the Optimal Homotopy Perturbation Method," Mathematics, MDPI, vol. 11(14), pages 1-22, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:14:p:3078-:d:1192544
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    References listed on IDEAS

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    1. Wang, Xuezhen & Zhang, Huasheng, 2023. "Intelligent control of convergence rate of impulsive dynamic systems affected by nonlinear disturbances under stabilizing impulses and its application in Chua’s circuit," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    2. Constantin Bota & Bogdan Căruntu & Dumitru Ţucu & Marioara Lăpădat & Mădălina Sofia Paşca, 2020. "A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order," Mathematics, MDPI, vol. 8(8), pages 1-12, August.
    3. Z. Ayati & A. Badiepour & Manuel De Le n, 2022. "Two New Modifications of the Exp-Function Method for Solving the Fractional-Order Hirota-Satsuma Coupled KdV," Advances in Mathematical Physics, Hindawi, vol. 2022, pages 1-12, July.
    4. Rafiq, Muhammad Hamza & Raza, Nauman & Jhangeer, Adil, 2023. "Dynamic study of bifurcation, chaotic behavior and multi-soliton profiles for the system of shallow water wave equations with their stability," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
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    Cited by:

    1. Remus-Daniel Ene & Nicolina Pop, 2023. "Approximate Closed-Form Solutions for a Class of 3D Dynamical Systems Involving a Hamilton–Poisson Part," Mathematics, MDPI, vol. 11(23), pages 1-26, November.

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