Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form

Abstract

We present an efficient numerical method for computing Fourier-Taylor expansions of stable/unstable manifolds associated with hyperbolic periodic orbits. Three features of the method are (1) that we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) that the method admits natural a-posteriori error analysis, and (3) that the method does not require numerical integrating the vector field. Our method is based on the Parameterization Method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the (un)stable manifold. The method requires only that some mild non-resonance conditions hold between the Floquet multipliers of the periodic orbit. The novelty of the the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier-Taylor expansion. We present a number of example computations, including stable/unstable manifolds in phase space dimension as hight as ten, computation of manifolds which are two and three dimensional, and computation of some homoclinic connecting orbits.

Computing (Un)stable Manifolds with Validated Error Bounds: Non-resonant and Resonant Spectra

“…Since closed-form analytic expressions for stable/unstable manifolds are rarely available, considerable effort goes into developing numerical techniques for their approximation, see e.g., Krauskopf et al (2005), Haro and da la Llave (2006), Beyn and Kless (1998) and Castelli et al (2015) and the references therein. One powerful tool for studying invariant manifolds (stable, unstable, strongly (un)stable and even more general ones) is the parameterization method of Cabré et al (2003aCabré et al ( , b, 2005.…”

“…The parameterization method is based on formulating certain operator equations (invariance equations) which simultaneously describe both the dynamics on the manifold and its embedding. The method has been implemented numerically to study a variety of problems involving stable and unstable manifolds of equilibria and fixed points (Mireles-James 2013; Mireles-James and Mischaikow 2013; Mireles James and Lomelí 2010 ;Haro 2011;van den Berg et al 2011), stable/unstable manifolds of periodic orbits for differential equations (Guillamon and Huguet 2009;Huguet and de la Llave 2013;Castelli et al 2015), quasiperiodic invariant sets in dynamical systems (Haro and da la Llave 2006) and more recently in order to simultaneously compute invariant manifolds with their unknown dynamics Haro et al 2014), to mention just a few examples.…”

“…The parameterization method has already been adapted to this context. See Guillamon andHuguet (2009), Huguet andde la Llave (2013) and Castelli et al (2015) for more details and numerical examples. Using the techniques of the present work it should be possible to validate these computations even in the presence of a resonance between the Floquet multipliers.…”

Analytic enclosure of the fundamental matrix solution

“…The main motivation behind our investigation is our interest in validated computation of analytic parameterizations of stable and unstable manifolds of hyperbolic periodic orbits of vector fields, as presented in [6]. The theoretical foundation of the parameterization method can be found in [1], [2], [3].…”

“…The theoretical foundation of the parameterization method can be found in [1], [2], [3]. The ingredients necessary for computer-assisted validation of the methods of [6] are the analytic representations of both the orbit and the tangent bundle. The latter can be accomplished via analytic representation of the fundamental matrix solution.…”