Reconnection of unstable/stable manifolds of the Harper map - Asymptotics-beyond-all-orders approach

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 = -∞ and is analytically continued to the other sectors, where the solution acquires new terms describing heteroclinic tangles. It is shown that when the parameter is changed, upon reaching the reconnection threshold, the unstable/stable manifolds acquire new oscillatory parts corresponding to the heteroclinic tangle after the reconnection.