A new fast multiple-shooting method for computing periodic orbits in symplectic maps leveraging simultaneous Floquet vector computation to avoid large linear systems

Abstract

Given a 4D symplectic map $F_0$ that has a normally hyperbolic invariant cylinder foliated by invariant tori, those with rational rotation numbers are themselves foliated by subharmonic periodic orbits (SPOs). If $F_0$ is part of a perturbative family $F_\varepsilon$, one is often interested in computing those SPOs which persist for $\varepsilon >0$. Assuming that a persisting SPO of $F_0$ has been identified, in this paper, we develop a quasi-Newton method which solves for the SPO simultaneously with its Floquet vectors and multipliers. This in turn enables continuation by the perturbation parameter $\varepsilon$. The resulting SPO and Floquet vectors are then used to compute Taylor parameterizations of the SPO's weak stable and unstable manifolds, if they exist. Our quasi-Newton method is based on an adaptation of the parameterization method for invariant tori, with this paper being the first-ever to apply such a framework to directly compute periodic orbit points themselves. The new algorithm improves on efficiency compared to prior multi-shooting methods for SPOs, and notably applies to the case of stroboscopic maps of 2.5 DOF Hamiltonian flows resulting from periodic perturbations of 2 DOF systems. The tools have been successfully used for studies of resonant orbits in perturbed real-life celestial systems, the results of which are summarized as a demonstration of the methods' utility.