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

Bhanu Kumar

TL;DR

The paper tackles the challenge of computing long subharmonic periodic orbits (SPOs) and their Floquet data in perturbative 4D symplectic maps arising from 2.5 DOF Hamiltonian systems. It introduces a fast quasi-Newton method that solves for SPO points and Floquet directions/multipliers simultaneously, leveraging a parameterization-method framework to decouple corrections and avoid large linear systems. The method yields not only SPOs but also constant, easily usable Floquet data, enabling high-order Taylor parameterizations of weak separatrices and subsequent separatrix analysis within normally hyperbolic invariant manifolds. Demonstrations on stroboscopic maps from planar PCR3BP CCR4BP models show efficient continuation of SPOs across perturbations and reveal resonance-driven separatrix intersections that account for torus destruction. The results have practical significance for celestial mechanics, enabling robust, scalable analysis of resonant orbits and their global impact on transport and chaos in multi-body 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.

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

TL;DR

The paper tackles the challenge of computing long subharmonic periodic orbits (SPOs) and their Floquet data in perturbative 4D symplectic maps arising from 2.5 DOF Hamiltonian systems. It introduces a fast quasi-Newton method that solves for SPO points and Floquet directions/multipliers simultaneously, leveraging a parameterization-method framework to decouple corrections and avoid large linear systems. The method yields not only SPOs but also constant, easily usable Floquet data, enabling high-order Taylor parameterizations of weak separatrices and subsequent separatrix analysis within normally hyperbolic invariant manifolds. Demonstrations on stroboscopic maps from planar PCR3BP CCR4BP models show efficient continuation of SPOs across perturbations and reveal resonance-driven separatrix intersections that account for torus destruction. The results have practical significance for celestial mechanics, enabling robust, scalable analysis of resonant orbits and their global impact on transport and chaos in multi-body systems.

Abstract

Given a 4D symplectic map 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 is part of a perturbative family , one is often interested in computing those SPOs which persist for . Assuming that a persisting SPO of 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 . 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.
Paper Structure (39 sections, 4 theorems, 93 equations, 7 figures)

This paper contains 39 sections, 4 theorems, 93 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem Setting and Statement
  3. Problem Setting
  4. The Problem: Computing Perturbed Subharmonic Periodic Orbits
  5. Motivation: Hamiltonian Flow Maps and Physical Models
  6. 2 DOF Hamiltonian Systems and NHIMs of Flow-Periodic Orbits
  7. Example: planar circular restricted 3-body problem (PCR3BP)
  8. Periodic Perturbations of 2 DOF Systems: the 2.5 DOF Case
  9. Example: planar concentric circular restricted 4-body problem
  10. Stroboscopic Maps, NHIMs, Tori, and Subharmonic Periodic Orbits
  11. An Efficient Method for Computing Subharmonic Periodic Orbits and Floquet Vectors
  12. The Parameterization Method for Invariant Manifolds
  13. Parameterization Method-Style Equations for Subharmonic Periodic Orbits and Floquet Vectors
  14. Preliminaries: Solution Methods for Common Equations and Procedures
  15. Fixed-point iteration for equations of form $\lambda_{a}(k) u(k) - \lambda_{b}(k) u(k+1 \mod q) = b(k)$
  16. ...and 24 more sections

Key Result

Lemma 1

If $0<C <1$, $A$ is a contraction map. Hence, in this case the iteration $u_{n+1} = A(u_{n})$ uniformly converges exponentially fast as $n \rightarrow \infty$ to the length-$q$ solution sequence $u$ of Eq. xi3contract (and thus also cohomHyp).

Figures (7)

  • Figure 1: (L) Jupiter-Ganymede PCRTBP 4:3 MMR unstable periodic orbits, (R) Orbit $\omega$ vs $x$-intercept plot kumar2023aas
  • Figure 2: Ganymede 4:3 MMR unstable orbits of CCR4BP map, $\varepsilon = 0.0, 8.0 \times 10^{-6}, 2.5265 \times 10^{-5}$ (from top to bottom) kumar2023aas
  • Figure 3: 4:3 Ganymede MMR 11/34 flow-SPO, including zoomed view on right, Jupiter-Europa-Ganymede CCR4BP
  • Figure 4: 4:3 Ganymede MMR 12/37 flow-SPO, including zoomed view on right, Jupiter-Europa-Ganymede CCR4BP kumar2023aas
  • Figure 5: Ganymede 4:3 MMR unstable secondary resonant periodic orbits and separatrices for stroboscopic map of physical-mass Jupiter-Europa-Ganymede CCR4BP. Plot in "action-angle-like" coordinates. Stable separatrices in blue, unstable in red.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Remark 1
  • Remark 2
  • Remark 3
  • Claim
  • proof
  • Remark 4
  • Claim
  • proof
  • Remark 5
  • Claim
  • ...and 10 more