The topology of nested homoclinic and heteroclinic tangles

Homoclinic and heteroclinic tangles are fundamental phase space structures that help in organizing the transport defined by a chaotic map on a two-dimensional phase space. Previous work introduced the technique of “homotopic lobe dynamics” to study the topology of a homoclinic tangle. The present work generalizes homotopic lobe dynamics to describe the topology of a network of nested homoclinic and/or heteroclinic tangles. Using this theory, we demonstrate that the topological analysis of a network of nested tangles can lead to an efficient symbolic description of a chaotic phase space, even in the vicinity of islands of stability. By increasing the amount of topological information used (e.g. by introducing additional tangles at finer scales), this technique suggests an efficient route for defining accurate symbolic dynamics for phase spaces that exhibit a mixture of chaos and regularity.