High-order parameterization of (un)stable manifolds for hybrid maps: Implementation and applications

• Published in: Communications in nonlinear science & numerical simulation 2017-12, Vol.53, p.184-201
• Main Authors: Naudot, Vincent, Mireles James, J.D., Lu, Qiuying
• Format: Article
• Language: English
• Subjects: Advection; Differential equations; Dynamical systems; Embedding; Hybrid control systems; Hybrid systems; Manifolds; Mathematical models; Numerical analysis; Parameterization
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2017.03.005

•We develop a numerical method for computing jets of the parameterization of the stable/unstable manifold of a fixed point of a certain kind of hybrid dynamical systems. These system consist of flow under a differential equation followed by a kick.•The parameterizations we compute can follow folds in the embedding and recover the dynamics on the manifold.•We prove that our algorithm computes the jets of the desired invariant object.•We implement our algorithm for a particular hybrid dynamical system and give numerical evidence for a topological horseshoe. In this work we study, from a numerical point of view, the (un)stable manifolds of a certain class of dynamical systems called hybrid maps. The dynamics of these systems are generated by a two stage procedure: the first stage is continuous time advection under a given vector field, the second stage is discrete time advection under a given diffeomorphism. Such hybrid systems model physical processes where a differential equation is occasionally kicked by a strong disturbance. We propose a numerical method for computing local (un)stable manifolds, which leads to high order polynomial parameterization of the embedding. The parameterization of the invariant manifold is not the graph of a function and can follow folds in the embedding. Moreover we obtain a representation of the dynamics on the manifold in terms of a simple conjugacy relation. We illustrate the utility of the method by studying a planar example system.