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 reconnectionof invariant manifolds. The method of asymptotics beyond all orders (ABAO) isused to construct unstable/stable manifolds of the Harper map. By enlarging theneighborhood of a singularity, the perturbative solution of the unstablemanifold is expressed as a Borel summable asymptotic expansion in a sectorincluding $t=-\infty$ and is analytically continued to the other sectors, wherethe solution acquires new terms describing heteroclinic tangles. When theparameter changes to the reconnection threshold, the unstable/stable manifoldsare shown to acquire new oscillatory portion corresponding to the heteroclinictangle after the reconnection.