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 $A:t\mapsto A(t)$ 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.