On GIT quotients of the symplectic group, stability and bifurcations of periodic orbits (with a view towards practical applications)

Abstract

In this article, we will introduce a collection of tools aimed at studying periodic orbits of Hamiltonian systems, their (linear) stability, and their bifurcations. We will provide topological obstructions to the existence of orbit cylinders of symmetric orbits, for mechanical systems preserved by anti-symplectic involutions (e.g. the circular restricted three-body problem). Such cylinders induce continuous paths which do not cross the bifurcation locus of suitable GIT quotients of the symplectic group, which are branched manifolds whose topology provide the desired obstructions. Namely, the complement of the corresponding loci consist of several connected components which we enumerate and explicitly describe; by construction these cannot be joined by a path induced by an orbit cylinder. We also provide preferred normal forms for each regular and singular component. We further introduce a notion of signature for symmetric orbits, which extends the notion from Krein theory (which only applies for elliptic orbits), to allow also for the case of symmetric orbits which are hyperbolic. This signature helps predict at which points of a symmetric orbit a bifurcation arises. This gives a general theoretical framework for the study of stability and bifurcations of symmetric orbits, with a view towards practical and numerical implementations within the context of space mission design. This is the subject of the follow-up paper $\href{https://doi.org/10.1007/978-3-319-72278-8}{[7]}$, where this framework is supported by numerical work.