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Homoclinic orbits to invariant tori in Hamiltonian systems. (English) Zbl 1140.37355

Jones, Christopher K.R.T. (ed.) et al., Multiple-time-scale dynamical systems. Proceedings of the IMA workshop, Minneapolis, MN, USA, 1997-98. New York, NY: Springer (ISBN 0-387-95126-1/hbk). IMA Vol. Math. Appl. 122, 1-27 (2001).
Summary: We consider a perturbation of an integrable Hamiltonian system which possesses invariant tori with coincident whiskers (like some rotators and a pendulum). Our goal is to measure the splitting distance between the perturbed whiskers. Emphasis is put on the detection of their intersections, which give rise to homoclinic orbits to the perturbed tori.
A geometric method is presented which takes into account the Lagrangian properties of the whiskered tori. In this way, the splitting distance is the gradient of an splitting potential. In the regular case (also known as a priori-unstable: the Lyapunov exponents of the whiskered tori remain fixed), the splitting potential is well-approximated by a Melnikov potential. This method is designed for the singular case (also known as a priori-stable: the Liapunov exponents of the whiskered tori approach to zero when the perturbation tends to zero).
For the entire collection see [Zbl 0952.00030].

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
70H08 Nearly integrable Hamiltonian systems, KAM theory
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