Detection of Phase Space Structures of the Cat Map with Lagrangian Descriptors

Abstract

The goal of this paper is to apply Lagrangian Descriptors (LDs), a technique based on Dynamical Systems Theory (DST) to reveal the phase space structures present in the well-known Arnold’s cat map. This discrete dynamical system, which represents a classical example of an Anosov diffeomorphism that is strongly mixing, will provide us with a benchmark model to test the performance of LDs and their capability to detect fixed points, periodic orbits and their stable and unstable manifolds present in chaotic maps. In this work we show, both from a theoretical and a numerical perspective, how LDs reveal the invariant manifolds of the periodic orbits of the cat map. The application of this methodology in this setting clearly illustrates the chaotic behavior of the cat map and highlights some technical numerical difficulties that arise in the identification of its phase space structures.