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Approximate solutions of a class of complex nonlinear dynamical systems

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  • Mahmoud, Gamal M.

Abstract

Nonlinear dynamical systems, being a realistic representation of nature, often exhibit a somewhat complicated behaviour. Their analysis requires a thorough investigation into the solutions of the governing nonlinear differential equations. In this paper, an approximate method is presented for solving complex nonlinear differential equations of the form: z̈+ω2z+εf(z,z̄,ż,z̄̇)=0,where z is a complex function and ε is a small parameter. It is based on the generalized averaging method which we have developed recently. Our approach can be viewed as a generalization of the approximate method based on the Krylov–Bogoliubov averaging method. The study of these systems is of interest to several fields of statistical mechanics, physics, electronics and engineering. Application of this method to special cases is performed for the purpose of comparison with numerical computations. Excellent agreement is found for reasonably large values of ε, which shows the applicability of this method to this kind of nonlinear dynamical systems. This agreement gives extra confidence that the analytical results are correct. These analytical results can be used as a theoretical guidance for doing further numerical or theoretical studies.

Suggested Citation

  • Mahmoud, Gamal M., 1998. "Approximate solutions of a class of complex nonlinear dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 253(1), pages 211-222.
    • Handle: RePEc:eee:phsmap:v:253:y:1998:i:1:p:211-222
    DOI: 10.1016/S0378-4371(98)00041-7
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    References listed on IDEAS

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    1. Mahmoud, Gamal M., 1993. "On the generalized averaging method of a class of strongly nonlinear forced oscillators," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 199(1), pages 87-95.
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    Citations

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    Cited by:

    1. Cveticanin, L., 2003. "Analytic solution of the system of two coupled differential equations with the fifth-order non-linearity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 317(1), pages 83-94.
    2. Li, Wei & Li, Jiaorui & Chen, Weisheng, 2012. "The reliability of a stochastically complex dynamical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(13), pages 3556-3565.
    3. Li, Wei & Xu, Wei & Zhao, Junfeng & Wu, Haibo, 2007. "The study on stationary solution of a stochastically complex dynamical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 385(2), pages 465-472.
    4. Mahmoud, Gamal M. & Aly, Shaban A.H., 2000. "On periodic solutions of parametrically excited complex non-linear dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 278(3), pages 390-404.
    5. Weaam Alhejaili & Alvaro H. Salas & Samir A. El-Tantawy, 2022. "Analytical and Numerical Study on Forced and Damped Complex Duffing Oscillators," Mathematics, MDPI, vol. 10(23), pages 1-13, November.
    6. Xu, Yong & Xu, Wei & Mahmoud, Gamal M, 2004. "On a complex beam–beam interaction model with random forcing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(3), pages 347-360.
    7. Weaam Alhejaili & Alvaro H. Salas & Samir A. El-Tantawy, 2023. "Ansatz and Averaging Methods for Modeling the (Un)Conserved Complex Duffing Oscillators," Mathematics, MDPI, vol. 11(9), pages 1-12, April.
    8. Xu, Yong & Xu, Wei & Mahmoud, Gamal M., 2008. "On a complex Duffing system with random excitation," Chaos, Solitons & Fractals, Elsevier, vol. 35(1), pages 126-132.
    9. Cveticanin, L., 2001. "Analytic approach for the solution of the complex-valued strong non-linear differential equation of Duffing type," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 297(3), pages 348-360.
    10. Xu, Yong & Zhang, Huiqing & Xu, Wei, 2007. "On stochastic complex beam–beam interaction models with Gaussian colored noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 384(2), pages 259-272.
    11. M. Mahmoud, Gamal & A. Mohamed, Ahmed & A. Aly, Shaban, 2001. "Strange attractors and chaos control in periodically forced complex Duffing's oscillators," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 292(1), pages 193-206.

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