Deep Learning of Conjugate Mappings

Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincaré first made this connection by through what is now known as a Poincaré map. Unfortunately, in all but the simplest systems, an explicit form for such a mapping remains outstanding. In this video I propose a method for obtaining explicit Poincaré mappings by leveraging deep learning to construct an invertible coordinate transformation into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is based on an autoencoder, which allows for dimensionality reduction, and has the advantage of classifying chaotic systems using the equivalence relation of topological conjugacies. ArXiv link: GitHub link: Deep Learning of Conjugate Mappings Jason J. Bramburger, Steven L. Brunto