Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 585 - 602, Volume 9, Issue 3, May 2003      doi:10.3934/dcds.2003.9.585

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Karsten Matthies - Freie Universität Berlin, Institut für Mathematik I, Arnimallee 2-6, 14195 Berlin, Germany (email)

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 $\epsilon \rightarrow 0$.

Keywords:  Homoclinic orbit, semilinear parabolic equations, splitting, exponential smallness.
Mathematics Subject Classification:  37L99, 37C29, 37G20.

Received: November 2001;      Revised: December 2002;      Available Online: February 2003.