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Strange attractors and chaos control in periodically forced complex Duffing's oscillators

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  • M. Mahmoud, Gamal
  • A. Mohamed, Ahmed
  • A. Aly, Shaban

Abstract

An interesting and challenging research subject in the field of nonlinear dynamics is the study of chaotic behavior in systems of more than two degrees of freedom. In this work we study fixed points, strange attractors, chaotic behavior and the problem of chaos control for complex Duffing's oscillators which represent periodically forced systems of two degrees of freedom. We produce plots of Poincaré map and study the fixed points and strange attractors of our oscillators. The presence of chaotic behavior in these models is verified by the existence of positive maximal Lyapunov exponent. We also calculate the power spectrum and consider its implications regarding the properties of the dynamics. The problem of controlling chaos for these oscillators is studied using a method introduced by Pyragas (Phys. Lett. A 170 (1992) 421), which is based on the construction of a special form of a time-continuous perturbation. The study of coupled periodically forced oscillators is of interest to several fields of physics, mechanics and engineering. The connection of our oscillators to the nonlinear Schrödinger equation is discussed.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:292:y:2001:i:1:p:193-206
    DOI: 10.1016/S0378-4371(00)00590-2
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    References listed on IDEAS

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    1. 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.
    2. 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.
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    Cited by:

    1. Mahmoud, Gamal M. & Mahmoud, Emad E. & Farghaly, Ahmed A. & Aly, Shaban A., 2009. "Chaotic synchronization of two complex nonlinear oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2858-2864.
    2. Zhang, Jian-Gang & Li, Xian-Feng & Chu, Yan-Dong & Yu, Jian-Ning & Chang, Ying-Xiang, 2009. "Hopf bifurcations, Lyapunov exponents and control of chaos for a class of centrifugal flywheel governor system," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2150-2168.
    3. 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.
    4. 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.
    5. Gao, Xin & Yu, Juebang, 2005. "Chaos in the fractional order periodically forced complex Duffing’s oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 1097-1104.
    6. 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.
    7. Wang, Ruiqi & Deng, Jin & Jing, Zhujun, 2006. "Chaos control in duffing system," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 249-257.
    8. Sachin Bhalekar, 2013. "Infinite-Scroll Attractor Generated by the Complex Pendulum Model," International Journal of Analysis, Hindawi, vol. 2013, pages 1-3, March.

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