# Chebyshev-Taylor parameterization of stable/unstable manifolds for periodic orbits: implementation and applications

## TL;DR

This paper introduces a high-order Chebyshev-Taylor series method combined with the parameterization approach to accurately compute stable and unstable manifolds of periodic orbits in dynamical systems, with applications to celestial mechanics and chaos theory.

## Contribution

It develops a recursive linear differential equation framework and spectral boundary value problem solutions for invariant manifolds, enhancing accuracy and robustness over previous methods.

## Key findings

- Effective for Lorenz system analysis
- Successful in celestial three-body problems
- Able to compute cycle-to-cycle connecting orbits

## Abstract

This paper develops a computational method for studying stable/unstable manifolds attached to periodic orbits of differential equations. The method uses high order Chebyshev-Taylor series approximations in conjunction with the parameterization method -- a general functional analytic framework for invariant manifolds. The parameterization method 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. The key to the approach is the derivation of a recursive system of linear differential equations describing the Taylor coefficients of the invariant manifold. We find periodic solutions of these equations by solving a coupled collection of boundary value problems with Chebyshev spectral methods. We discuss the performance of the method for the Lorenz system, and for 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.