Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing

Karsten Matthies

doi:10.3934/dcds.2003.9.585

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2003, Volume 9, Issue 3: 585-602. Doi: 10.3934/dcds.2003.9.585

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Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing

Karsten Matthies^1,

1.
Freie Universität Berlin, Institut für Mathematik I, Arnimallee 2-6, 14195 Berlin

Received: November 2001

Revised: December 2002

Published: May 2003

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Abstract

Homoclinic orbits of semilinear parabolic partial differential equations can split under time-periodic forcing as for ordinary differential equations. The stable and unstable manifold may intersect transverse at persisting homoclinic points. The size of the splitting is estimated to be exponentially small of order exp$(-c/\epsilon)$ in the period $\epsilon$ of the forcing with $\epsilon \rightarrow 0$.

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Mathematics Subject Classification: 37L99, 37C29, 37G20.

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