# On bifurcation of eigenvalues along convex symplectic paths

### Yinshan Chang

College of Mathematics, Sichuan University, Chengdu 610065, China### Yiming Long

Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China### Jian Wang

Chern Institute of Mathematics, Nankai University, Tianjin 300071, China, IMPA, Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, Brazil

## Abstract

We consider a continuously differentiable curve $t↦γ(t)$ in the space of $2n×2n$ real symplectic matrices, which is the solution of the following ODE:

where \( J = J_{2n} = \limits^{\text{def}}\left[\begin{matrix} 0 & \mathrm{Id}_{n} \\ −\mathrm{Id}_{n} & 0 \\ \end{matrix}\right] \) and $A:t↦A(t)$ is a continuous path in the space of $2n×2n$ real matrices which are symmetric. Under a certain convexity assumption (which includes the particular case that $A(t)$ is strictly positive definite for all $t∈R$), we investigate the dynamics of the eigenvalues of $γ(t)$ 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: ${t∈R:γ(t)has a Krein indefinite eigenvalue of modulus1}$ is a discrete set.

## Cite this article

Yinshan Chang, Yiming Long, Jian Wang, On bifurcation of eigenvalues along convex symplectic paths. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 1, pp. 75–102

DOI 10.1016/J.ANIHPC.2018.04.001