Table of Links

• Abstract
• Introduction
• Preliminaries
• The B-signature
• GIT sequence: low dimensions
• GIT sequence: arbitrary dimension
• Appendix A. Stability, the Krein–Moser theorem, and refinements and References

1. Introduction

The stability of periodic orbits is a central topic in the study of Hamiltonian systems, going back to the problem of stability of the solar system in celestial mechanics. Ubiquitous in the study of ODEs, the notion of stability arises whenever studying orbits in families and their bifurcations, a practice which entails both theoretical and practical interest. For instance, from the perspective of space mission design, orbits used for parking a spacecraft around a target Moon should be as stable as possible, in order to minimize fuel corrections and station-keeping. From a mathematical point of view, the key notions of stability of a system come in three flavors, related by the following implications:

Non-linear (Lyapunov) stability ⇒ linear stability ⇒ spectral stability.

Non-linear stability, roughly speaking, means that trajectories which start near a given periodic orbit stay near the orbit for all time. Linear stability corresponds to stability of the origin for the linearized dynamics, i.e. the orbits of the linearized system should stay bounded. For a Hamiltonian system, this means that the eigenvalues of the monodromy matrix of the corresponding orbit should lie in the unit circle, and be semi-simple. Spectral stability, on the other hand, requires that eigenvalues all lie in the unit circle, but allows them to have multiplicity (so that orbits can escape to infinity in polynomial time, rather than exponential). In this paper, we will focus on the notion of linear stability.

In the presence of symmetry, the study of linear stability of periodic orbits which are preserved by the symmetry can be significantly refined. With this end in mind, the first author and Urs Frauenfelder introduced in [FM] the notion of the GIT sequence, as a refinement of the Broucke stability diagram [Br69], via the notion of B-signature. The GIT sequence consists of a sequence of three spaces and maps between them whose topology encodes stability and bifurcations of periodic orbits, as well as their eigenvalue configurations, and provides obstructions to the existence of regular cylinder of orbits. In low dimensions, the spaces can be visualized in the plane or in three-dimensional space, which makes them amenable for numerical work. We should note that while the GIT sequence is designed to study linear stability, it blurs its distinction with spectral stability.

Indeed, recall that the Krein–Moser theorem gives a criterion for when a Krein biurcation may occur (i.e. two elliptic eigenvalues of the monodromy matrix come together and then bifurcate out of the circle). Our refinement gives a similar criterion for the situation when two hyperbolic eigenvalues come together at a hyperbolic eigenvalue of multiplicity two and then become complex, but for the case of symmetric orbits. We call such a transition a HN -transition, and the high-multiplicity eigenvalue, the transit eigenvalue. Whether or not such a transition may occur is completely determined by the B-signature of the transit eigenvalue. Namely, the following result is a consequence of our topological study of the symplectic group.

Theorem A. Consider a Hamiltonian with arbitrary degrees of freedom, admitting a symmetry. Let t 7→ γt , t ∈ [0, 1], be a family of symmetric periodic orbits, undergoing an HN -transition. Then the B-signature of the transit eigenvalue is indefinite.

The definition of B-signature will be given in Section 3, and the proof of this theorem is obtained in Appendix A.

Acknowledgements. The authors are grateful to Urs Frauenfelder, whose ideas inspired this paper. A. Moreno is currently supported by the Sonderforschungsbereich TRR 191 Symplectic Structures in Geometry, Algebra and Dynamics, funded by the DFG (Projektnummer 281071066 – TRR 191), and also by the DFG under Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster).