Abstract
The Melnikov method was extended to perturbed planar non-Hamiltonian integrable systems with slowly-varying angle parameters. Based on the analysis of the geometric structure of unperturbed systems, the condition of transversely homoclinic intersection was established. The generalized Melnikov function of the perturbed system was presented by applying the theorem on the differentiability of ordinary differential equation solutions with respect to parameters. Chaos may occur in the system if the generalized Melnikov function has simple zeros.
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References
LIU Zeng-rong. The Melnikov method in researching chaos [A]. In: GUO Zhong-Heng Ed. Modern Mathematics and Mechanics [C]. Beijing: Peking University Press, 1987,269-290. (in Chinese)
Wiggins S. Global Bifurcations and Chaos[M]. Berlin: Springer-Verlag,1988.
Holmes P J. Averaging and chaotic motions in forced oscillations[J]. SIAM J Appl Math,1980,38(1): 65-80;1980,40(1):167–168.
JIANG Ji-fa, LIU Zeng-rong. Subharmonic bifurcation and horseshoe in a non-Hamiltonian system[J]. Math Appl Sinica, 1987,10(4):504-508. (in Chinese)
CHEN Li-qun, LIU Yan-zhu. Chaos in quasiperiodically perturbed planar non-Hamiltonian integrable systems[J]. J Shanghai Jiaotong Univ, 1996,30(11):28-31. (in Chinese)
CHEN Li-qun. The conceptual evolution of chaos in science[J]. Nature J, 1991,14(7):619-624. (in Chinese)
Wiggins S. Normally Hyperbolic Invariant Manifolds in Dynamical Systems[M]. Berlin: Springer-Verlag, 1994.
Hale J. Ordinary Differential Equations[M]. London: Robert E Krieger,1980.
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Chen, Lq. Chaos in Perturbed Planar Non-Hamiltonian Integrable Systems with Slowly-Varying Angle Parameters. Applied Mathematics and Mechanics 22, 1301–1305 (2001). https://doi.org/10.1023/A:1016378223861
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DOI: https://doi.org/10.1023/A:1016378223861