On bifurcation of eigenvalues along convex symplectic paths

TL;DR

This paper studies how eigenvalues of symplectic matrices evolve under a differential equation, providing new insights into their bifurcation behavior and stability in Hamiltonian systems, extending classical theorems.

Contribution

It generalizes the Krein-Lyubarskii theorem by analyzing eigenvalue bifurcations for symplectic paths under convexity assumptions, including explicit asymptotics.

Findings

Eigenvalues exhibit specific bifurcation patterns under convexity conditions.

First order asymptotics of eigenvalues are explicitly derived.

The set of parameters where indefinite eigenvalues occur is discrete.

Abstract

We consider a continuously differentiable curve $t\mapsto \gamma(t)$ in the space of $2n\times 2n$ real symplectic matrices, which is the solution of the following ODE: $\frac{\mathrm{d}\gamma}{\mathrm{d}t}(t)=J_{2n}A(t)\gamma(t), \gamma(0)\in\operatorname{Sp}(2n,\mathbb{R})$, where $J=J_{2n}\overset{\text{def}}{=}\begin{bmatrix}0 & \operatorname{Id}_n\\-\operatorname{Id}_n & 0\end{bmatrix}$ and $A:t\mapsto A(t)$ is a continuous in the space of $2n\times2n$ real matrices which are symmetric. Under certain convexity assumption (which includes the particular case that $A(t)$ is strictly positive definite for all $t\in\mathbb{R}$), we investigate the dynamics of the eigenvalues of $\gamma(t)$ when $t$ varies, which are closely related to the stability of such Hamiltonian dynamical systems. We rigorously prove the qualitative behavior of the branching of eigenvalues and explicitly give the first order asymptotics of the eigenvalues. This generalizes classical Krein-Lyubarskii theorem on the analytic bifurcation of the Floquet multipliers under a linear perturbation of the Hamiltonian. As a corollary, we give a rigorous proof of the following statement of Ekeland: $\{t\in\mathbb{R}:\gamma(t)\text{ has a Krein indefinite eigenvalue of modulus }1\}$ is a discrete set.