Multi-Shooting Parameterization Methods for Invariant Manifolds and Heteroclinics of 2 DOF Hamiltonian Poincaré Maps, with Applications to Celestial Resonant Dynamics

Abstract Studying 2 degree-of-freedom (DOF) Hamiltonian dynamical systems often involves the computation of stable and unstable manifolds of periodic orbits, due to the homoclinic and heteroclinic connections they can generate. Such study is generally facilitated by the use of a Poincaré section, on which the manifolds form 1D curves. A common method of computing such manifolds in the literature involves linear approximations of the manifolds, while the author’s past work has developed a nonlinear manifold computation method best suited for the case of a single intersection between the periodic orbit and the chosen Poincaré section. However, linear manifold approximations may require large amounts of numerical integration for globalization, while the single-intersection assumption of the previous nonlinear method often does not hold. In the latter case, a full search for homoclinic or heteroclinic connections requires computing multiple manifold curves on the section for each orbit, which the previous method does not yield. In this paper, a parameterization method is developed and implemented for computing all the aforementioned multiple stable and unstable manifold curves in the case of multiple periodic orbit intersections with a chosen Poincaré section. The parameterization method developed avoids the need to compose polynomials with Poincaré maps—a requirement of some previous related algorithms—by using an intermediate step involving fixed-time maps. The step yields curves near the chosen Poincaré section lying on the flow’s periodic orbit manifolds, which are used to parameterize and compute the Poincaré map manifold curves themselves. These last curves and parameterizations in turn enable highly accurate computation of heteroclinics between periodic orbits. The method has already been used for various studies of resonant dynamics in the planar circular restricted 3-body problem, which are briefly summarized in this paper, demonstrating the algorithm’s utility for real-world investigations.