# Second order asymptotics for Krein indefinite multipliers with multiplicity two

## TL;DR

This paper derives second order asymptotic expansions for eigenvalues bifurcating from non-real Krein indefinite eigenvalues with multiplicity two in linear Hamiltonian systems in four dimensions.

## Contribution

It provides a detailed second order asymptotic analysis of eigenvalues near non-real Krein indefinite eigenvalues with multiplicity two.

## Key findings

- Second order asymptotics for eigenvalues bifurcating from non-real Krein indefinite eigenvalues.
- Analysis applies to 4x4 Hamiltonian systems with symmetric, continuously varying matrices.
- Results enhance understanding of spectral bifurcations in Hamiltonian dynamics.

## Abstract

We consider linear Hamiltonian equations in $\mathbb{R}^{4}$ of the following type \begin{equation} \frac{\mathrm{d}\gamma}{\mathrm{d}t}(t)=J_{4}A(t)\gamma(t), \gamma(0)\in\operatorname{Sp}(4,\mathbb{R}), \end{equation} where $J=J_{4}\overset{\text{def}}{=}\begin{bmatrix}0 & \operatorname{Id}_2\\-\operatorname{Id}_2 & 0\end{bmatrix}$ and $A:t\mapsto A(t)$ is a $C^1$-continuous curve in the space of $4\times 4$ real matrices which are symmetric. We obtain second order asymptotics for the eigenvalues bifurcated from non-real Krein indefinite eigenvalues with multiplicity two.