Studying network of symmetric periodic orbit families of the Hill problem via symplectic invariants

TL;DR

This paper uses symplectic invariants to analyze the network structure of symmetric periodic orbit families in the spatial circular Hill three-body problem, revealing their interconnectedness and bifurcation properties.

Contribution

It introduces a novel application of symplectic invariants and Conley-Zehnder indices to organize and understand the network of periodic orbit families in the Hill problem.

Findings

Construction of bifurcation graphs illustrating interconnected orbit families

Demonstration of the importance of symmetries in orbit interactions

Identification of the role of Conley-Zehnder index in bifurcation analysis

Abstract

In the framework of the spatial circular Hill three-body problem we illustrate the application of symplectic invariants to analyze the network structure of symmetric periodic orbit families. The extensive collection of families within this problem constitutes a complex network, fundamentally comprising the so-called basic families of periodic solutions, including the orbits of the satellite $g$, $f$, the libration (Lyapunov) $a,c$, and collision $\mathcal B_0$ families. Since the Conley-Zehnder index leads to a grading on the local Floer homology and its Euler characteristics, a bifurcation invariant, the computation of those indices facilitates the construction of well-organized bifurcation graphs depicting the interconnectedness among families of periodic solutions. The critical importance of the symmetries of periodic solutions in comprehending the interaction among these families is demonstrated.