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Predictions of quasi-periodic and chaotic motions in nonlinear Hamiltonian systems

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  • Luo, Albert C.J.

Abstract

In this paper, the mechanism of chaotic motion in nonlinear Hamiltonian systems is discussed based on the KAM theory and resonance overlap criterion. The internal resonances and the corresponding chaotic motions are determined analytically for weak interactions. A numerical method based on the energy spectrum is presented for prediction of quasi-periodic and chaotic motions in nonlinear Hamiltonian systems. The presented numerical method can be applied to integrable, nonlinear Hamiltonian systems with many degrees of freedom. A 2-DOF integrable, nonlinear Hamiltonian system is investigated as an example for demonstration of the procedure to numerically determine the chaotic motion in nonlinear Hamiltonian systems. Finally, the Poincare mapping surfaces of chaotic motions for such nonlinear Hamiltonian systems are illustrated. The phase planes, displacement surfaces (or potential domains), and the velocity surfaces (or kinetic energy domains) for the chaotic and quasi-periodic motions are illustrated. The analytical estimates of regular and chaotic motions in nonlinear Hamiltonian systems need to be further investigated. The mathematical theory should be developed for a better prediction of chaotic and quasi-periodic motions in nonlinear Hamiltonian systems with many degrees of freedom.

Suggested Citation

  • Luo, Albert C.J., 2006. "Predictions of quasi-periodic and chaotic motions in nonlinear Hamiltonian systems," Chaos, Solitons & Fractals, Elsevier, vol. 28(3), pages 627-649.
    • Handle: RePEc:eee:chsofr:v:28:y:2006:i:3:p:627-649
    DOI: 10.1016/j.chaos.2005.08.012
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    Cited by:

    1. Tigan, G. & Constantinescu, D., 2016. "Bifurcations in a family of Hamiltonian systems and associated nontwist cubic maps," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 128-135.
    2. Sun, Yeong-Jeu, 2009. "Robust tracking control of uncertain Duffing–Holmes control systems," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1282-1287.
    3. Sun, Yeong-Jeu, 2009. "An exponential observer for the generalized Rossler chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2457-2461.
    4. Sun, Yeong-Jeu, 2009. "A simple observer of the generalized Chen chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1641-1644.
    5. Sun, Yeong-Jeu, 2009. "On the state reconstructor design for a class of chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 815-820.
    6. Luo, Tong & Xu, Ming & Dong, Yunfeng, 2018. "Dynamics in the controlled center manifolds by Hamiltonian structure-preserving stabilization," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 149-158.
    7. Sun, Yeong-Jeu, 2009. "Solution bounds of generalized Lorenz chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 691-696.

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