A new fast multiple-shooting method for computing periodic orbits in symplectic maps leveraging simultaneous Floquet vector computation to avoid large linear systems
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911349962964992 |
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| author | Kumar, Bhanu |
| author_facet | Kumar, Bhanu |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_00149 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A new fast multiple-shooting method for computing periodic orbits in symplectic maps leveraging simultaneous Floquet vector computation to avoid large linear systems Kumar, Bhanu Dynamical Systems Earth and Planetary Astrophysics Chaotic Dynamics 37C25, 37C27, 37J12, 65P10, 70H12 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. |
| title | A new fast multiple-shooting method for computing periodic orbits in symplectic maps leveraging simultaneous Floquet vector computation to avoid large linear systems |
| topic | Dynamical Systems Earth and Planetary Astrophysics Chaotic Dynamics 37C25, 37C27, 37J12, 65P10, 70H12 |
| url | https://arxiv.org/abs/2601.00149 |