Second order asymptotics for Krein indefinite multipliers with multiplicity two

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$A:t\mapsto A(t)$isa$C^1$-continuouscurveinthespaceof$4\times 4$ real matrices which are symmetric. We obtain second order asymptotics for the eigenvalues bifurcated from non-real Krein indefinite eigenvalues with multiplicity two.