One-sided invariant manifolds, recursive folding, and curvature singularity in area-preserving nonlinear maps with nonuniform hyperbolic behavior

Abstract

Two-dimensional nonlinear models of conservative dynamics are typically nonuniformly hyperbolic in that there are nonhyperbolic trajectories that coexist with a “massive” hyperbolic region. We investigate the influence of nonhyperbolic points on the global geometric structure of a invariant manifolds associated with points of the hyperbolic region. As a case study, we consider a transformation of the Standard Map family and analyze the structure of invariant manifolds in the neighborhood of an isolated parabolic (fixed) point xp. This analysis shows the existence of lobes enclosing the parabolic point, that is, of simply connected regions containing xp whose boundary is formed by two continuous arcs of stable and unstable manifolds that intersect only at two points. From the existence of such regions, we derive that (i) there are points of the hyperbolic region where the local curvature of invariant manifolds is arbitrarily large and (ii) manifolds possess the recursively folding property. Property (ii) means that given an invariant manifold W and established an orientation on it, in the neighborhood of any point of the chaotic region there are nearby arcs of W that are traveled in opposite directions. We propose an archetypal model for which the existence of lobes and the recursive folding property can be derived analytically. The impact of nonuniform hyperbolicity on the evolution of physical processes that occur along with phase space mixing is also addressed.