Chebyshev–Taylor Parameterization of Stable/Unstable Manifolds for Periodic Orbits: Implementation and Applications

Abstract

This paper develops Chebyshev-Taylor spectral methods for studying stable/unstable manifolds attached to periodic solutions of differential equations. The work exploits the parameterization method -a general functional analytic framework for studying invariant manifolds. Useful features of the parameterization method include the fact that it can follow folds in the embedding, recovers the dynamics on the manifold through a simple conjugacy, and admits a natural notion of a-posteriori error analysis. Our approach begins by deriving a recursive system of linear differential equations describing the Taylor coefficients of the invariant manifold. We represent periodic solutions of these equations as solutions of coupled systems of boundary value problems. We discuss the implementation and performance of the method for the Lorenz system, and for the planar circular restricted three and four body problems. We also illustrate the use of the method as a tool for computing cycle-to-cycle connecting orbits.

High-order resonant orbit manifold expansions for mission design in the planar circular restricted 3-body problem

“…Note that our approach of using high order parameterizations of invariant manifolds to compute heteroclinic connections bears some similarity with prior studies; for instance, James and Murray [19] parameterized manifolds of periodic orbits using high order Chebyeshev-Taylor series, using the resulting 2D parameterizations to find connecting orbits. However, our study avoids dealing with 2D manifolds by using a Poincaré section to reduce the dimensionality of the problem, without sacrificing the accuracy which comes from using high order manifold expansions.…”

Resonant tori, transport barriers, and chaos in a vector field with a Neimark–Sacker bifurcation

“…The parameterization method is a general functional analytic framework for studying invariant manifolds of discrete and continuous time dynamical systems, first developed in [25,26,27] in the context of stable/unstable manifolds attached to fixed points of nonlinear mappings on Banach spaces, and later extended in [40,41,42] for studying whiskered tori. There is a thriving literature devoted to computational applications of the parameterization method, and the interested reader may want to consult [10,18,19,43,44,45,46,47,48,49,50,51,52,53,54], though the list is far from being exhaustive. A much more complete discussion is found in the book [55].…”