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Homoclinic orbits to invariant tori in Hamiltonian systems

Homoclinic orbits to invariant tori in Hamiltonian systems

Institute for Mathematics and Its Applications
**Amadeu Delshams**, Univ. Politécnica de
Catalunya

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 Liapunov 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), which is currently being researched.

This is a joint work with Pere Gutiérrez.