Heteroclinic Transition Motions in Periodic Perturbations of Conservative Systems with an Application to Forced Rigid Body Dynamics
Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan
We consider periodic perturbations of conservative systems. The unperturbed systems are assumed to have two nonhyperbolic equilibria connected by a heteroclinic orbit on each level set of conservative quantities. These equilibria construct two normally hyperbolic invariant manifolds in the unperturbed phase space, and by invariant manifold theory there exist two normally hyperbolic, locally invariant manifolds in the perturbed phase space. We extend Melnikovís method to give a condition under which the stable and unstable manifolds of these locally invariant manifolds intersect transversely. Moreover, when the locally invariant manifolds consist of nonhyperbolic periodic orbits, we show that there can exist heteroclinic orbits connecting periodic orbits near the unperturbed equilibria on distinct level sets. This behavior can occur even when the two unperturbed equilibria on each level set coincide and have a homoclinic orbit. In addition, it yields transition motions between neighborhoods of very distant periodic orbits, which are similar to Arnold diffusion for three or more degree of freedom Hamiltonian systems possessing a sequence of heteroclinic orbits to invariant tori, if there exists a sequence of heteroclinic orbits connecting periodic orbits successively.We illustrate our theory for rotational motions of a periodically forced rigid body. Numerical computations to support the theoretical results are also given.
heteroclinic motion, transition motion, chaos, conservative system, Melnikov method, rigid body
|Japan Society for the Promotion of Science
|This work was partially supported by the Japan Society for the Promotion of Science, Grantin-Aid for Scientific Research (C) (Subject No. 25400168).
Kazuyuki Yagasaki, “Heteroclinic Transition Motions in Periodic Perturbations of Conservative Systems with an Application to Forced Rigid Body Dynamics”, Regul. Chaotic Dyn., 23:4 (2018), 438–457
Citation in format AMSBIB
\by Kazuyuki Yagasaki
\paper Heteroclinic Transition Motions in Periodic Perturbations of Conservative Systems with an Application to Forced Rigid Body Dynamics
\jour Regul. Chaotic Dyn.
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