On bifurcation of eigenvalues along convex symplectic paths

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.