University of Helsinki, Helsinki 2006
Homoclinic Splitting without Trees
Doctoral dissertation, May 2006.
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.
This publication is copyrighted. You may download, display and print it for Your own personal use. Commercial use is prohibited.
© University of Helsinki 2006
Last updated 27.04.2006