Abstract

Homoclinic orbits play a crucial role in the dynamics of perturbations of the pendulum and sine-Gordon equations. In this paper we examine how well the homoclinic structures are preserved by symplectic discretizations. We discuss the property of exponentially small splitting distances between the stable and unstable manifolds for symplectic discretizations of the pendulum equation. A description of the sine-Gordon phase space in terms of the associated nonlinear spectral theory is provided. We examine how preservation of the homoclinic structures (i.e. the nonlinear spectrum) depends on the order of the accuracy and the symplectic property of the numerical scheme.

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