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Taylor Model-based Enclosure of Invariant Manifolds for Planar Diffeomorphisms and Applications

Abstract

Fundamental questions in the study of interesting dynamics of planar diffeomorphisms like the Henon map involve homoclinic phenomena, topological entropy and strange attractors. Inherently, answering these questions requires knowledge about the stable and unstable manifolds, which in the typical case in the plane are smooth curves. We present a method to find highly accurate Taylor Model enclosures of the invariant curves near hyperbolic fixed points. Successive iteration of these local enclosures yields similarly accurate enclosures of pieces of the global manifold tangle. Applications presented include the automatic computation of rigorous enclosures of all homoclinic points up to finite iterates. This allows to find symbolic dynamics in the original system and consequently compute rigorous lower bounds for its topological entropy.

J. Grote, M. Berz, K. Makino, S. Newhouse, Proc. Applied Mathematics and Mechanics 7 (2008) 1022907

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