Reconnection of Unstable/Stable Manifolds of the Harper Map

Details Reconnection of Unstable/Stable Manifolds of the Harper Map Reconnection of Unstable/Stable Manifolds of the Harper Map

2006-06-12

arxiv.org/abs/nlin/0606030v1

Nonlinear Sciences

Chaotic Dynamics

38 pages, 9 figures: revised version of nlin/0501042

Scientific paper

The Harper map is one of the simplest chaotic systems exhibiting reconnection of invariant manifolds. The method of asymptotics beyond all orders (ABAO) is used to construct unstable/stable manifolds of the Harper map. By enlarging the neighborhood of a singularity, the perturbative solution of the unstable manifold is expressed as a Borel summable asymptotic expansion in a sector including $t=-\infty$ and is analytically continued to the other sectors, where the solution acquires new terms describing heteroclinic tangles. When the parameter changes to the reconnection threshold, the unstable/stable manifolds are shown to acquire new oscillatory portion corresponding to the heteroclinic tangle after the reconnection.

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