Parameterized stable/unstable manifolds for periodic solutions of implicitly defined dynamical systems

• Published in: Chaos, solitons and fractals Vol. 161; p. 112345
• Main Authors: Timsina, Archana Neupane, Mireles James, J.D.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.08.2022
• Subjects: Computational methods; Implicitly defined dynamical systems; Invariant manifolds; Parameterization method; Periodic orbits; Computational methods; Periodic orbits; Parameterization method; Invariant manifolds; Implicitly defined dynamical systems
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2022.112345

We develop a multiple shooting parameterization method for studying stable/unstable manifolds attached to periodic orbits of systems whose dynamics is determined by an implicit rule. We represent the local invariant manifold using high order polynomials and show that the method leads to efficient numerical calculations. We implement the method for several example systems in dimension two and three. The resulting manifolds provide useful information about the orbit structure of the implicit system even in the case that the implicit relation is neither invertible nor single-valued. •That we develop a multiple shooting framework describing the local invariant stable/unstable manifolds attached to fixed points and periodic orbits of implicitly defined discrete time dynamical systems. The method generalizes the Parameterization method of Cabre, Fontich, and de la Llave and also earlier work of Lomeli and de la Llave.•We derive linear homological equations whose solutions are the jets of the parameterized local stable/unstable manifolds.•For several examples of implicitly defined discrete time dynamical systems we implement solvers for the homological equations, and compute to high order the Taylor expansions of the invariant manifolds. We illustrate the method for systems in dimension two and three, and for manifolds which are one and two dimensional.•We globalize the local invariant manifolds numerically, and by doing so locate homoclinic and heteroclinic connecting orbits for the implicitly defined example systems.