Combinatorics of linear stability for Hamiltonian systems in arbitrary dimension

Abstract

We address the general problem of studying linear stability and bifurcations of periodic orbits for Hamiltonian systems of arbitrary degrees of freedom. We study the topology of the GIT sequence introduced by the first author and Urs frauenfelder, in arbitrary dimension. In particular, we note that the combinatorics encoding the linear stability of periodic orbits is governed by a quotient of the associahedron. Our approach gives a topological/combinatorial proof of the classical Krein--Moser theorem, and refines it for the case of symmetric orbits.

Figures (10)

• Figure 1: The 2D GIT sequence. One obtains more refined information for symmetric orbits.
• Figure 2: The 3D Broucke stability diagram. Here, $\Gamma_{\pm 1}$ corresponds to eigenvalue $\pm 1$, $\Gamma_d$ to double eigenvalue, $\mathcal{E}^2$ to doubly elliptic (the stable region), and so on FM.
• Figure 3: The branches (represented as lines) are two-dimensional, and come together at the 1-dimensional "branching locus" (represented as points), where we cross from one region to another of the Broucke diagram. The $1$-dimensional loci collapse to points over each of the three singular points $(2,1),(0,-1), (-2,1) \in \mathbb R^2$.
• Figure 4: Bifurcations are encoded by a pencil of lines.
• Figure 5: The stability diagram for depressed cubics.

Theorems & Definitions (17)

• Theorem A
• Definition 2.1: GIT quotient
• Theorem 1: Wonenburger
• Definition 3.1: B/C-signs
• Theorem 2: FMb
• Definition 3.2: B/C-signature
• Proposition 4.1
• Definition 4.1
• Example 5.1
• Remark 5.2