University of Helsinki, Helsinki 2006

Homoclinic Splitting without Trees

Mikko Stenlund

Doctoral dissertation, May 2006.
University of Helsinki, Faculty of Science, Department of Mathematics and Statistics.

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, giving a simple construction of unstable KAM tori and their stable and unstable manifolds for analytic perturbations.

When the coupling takes place through an even trigonometric polynomial in the angle variables, we extend analytically the solutions of the equations of motion, order by order in the perturbation parameter, to a large neighbourhood of the real line representing time. Subsequently, we devise an asymptotic expansion for the splitting (matrix) associated with a homoclinic point. This expansion consists of contributions that are manifestly exponentially small in the limit of vanishing gravity, by a shift-of-countour argument. Hence, we infer a similar upper bound for the splitting itself.

In particular, the derivation of the result does not call for a tree expansion with explicit cancellation mechanisms.

The title page of the publication

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Last updated 27.04.2006