Symplectic methods in the numerical search of orbits in real-life planetary systems

Abstract

The intention of this article is to illustrate the use of invariants coming from Floer-type theories, as well as global topological methods, for practical purposes. Our intended audience is scientists interested in orbits of Hamiltonian systems (e.g. the three-body problem). In this paper, we illustrate the use of the GIT sequence introduced in [10] by the first and third authors, consisting of a sequence of spaces and maps between them. Roughly speaking, closed orbits of an arbitrary Hamiltonian system induce points in these spaces, and so their topology imposes restrictions on the existence of regular orbit cylinders, as well as encodes information on all types of bifurcations. We also consider the notion of the SFT-Euler characteristic, as the Euler characteristic of the local Floer homology groups associated to a closed orbit of a Hamiltonian system. This number stays invariant before and after a bifurcation, and can be computed in explicit terms from the eigenvalues of the reduced monodromy matrix. This makes it useful for practical applications, e.g. for predicting the existence of orbits for optimization and space mission design. For the case of symmetric orbits (i.e. preserved by an antisymplectic involution), which may be thought both as open or closed strings, we further consider the notion of the real Euler characteristic, as the Euler characteristic of the corresponding Lagrangian Floer homology group. We illustrate the practical use of these invariants, together with the GIT sequence, via numerical work based on the cell-mapping method as described in [16], for the Jupiter-Europa and the Saturn-Enceladus systems. These are currently systems of interest, as they fall in the agenda of space agencies like NASA, as these icy moons are considered candidates for harbouring conditions suitable for extraterrestrial life. CONTENTS 1. Introduction 2. The SFT-Euler characteristic of a periodic orbit 3. The real Euler characteristic of a symmetric orbit 4. Non-symmetric orbits and branching structure 5. Symmetric subtle division 6. Numerics Appendix A. Reduced vs nonreduced monodromy matrices Appendix B. Planar orbits and spatial vs. planar bifurcations Appendix C. Basis changes References

On doubly symmetric periodic orbits

“…Indeed, see e.g. [11,Section 6] for a numerical example of a planar-to-spatial period doubling bifurcation of doubly symmetric orbits.…”

“…We show that, in dimension four, doubly symmetric periodic orbits cannot be negative hyperbolic. This has a number of consequences: (1) all covers of doubly symmetric orbits are good, in the sense of Symplectic Field Theory [6]; (2) a non-degenerate doubly symmetric orbit is stable if and only if its CZ-index is odd; (3) a doubly symmetric orbit does not undergo period doubling bifurcation; and (4) there is always a stable orbit in any collection of doubly symmetric periodic orbits with negative SFT-Euler characteristic (as coined in [11]). The above results follow from:(5) a symmetric orbit is negative hyperbolic if and only its two B-signs (introduced in [10]) differ.…”