  This paper focuses on studying the bifurcation analysis of the eigenstructure
of the $\gamma$-parameterized generalized eigenvalue problem ($\gamma$-GEP)
arising in three-dimensional (3D) source-free Maxwell's equations with Pasteur
media, where $\gamma$ is the magnetoelectric chirality parameter. For the
weakly coupled case, namely, $\gamma < \gamma_{*} \equiv$ critical value, the
$\gamma$-GEP is positive definite, which has been well-studied by Chern et.\
al, 2015. For the strongly coupled case, namely, $\gamma > \gamma_{*}$, the
$\gamma$-GEP is no longer positive definite, introducing a totally different
and complicated structure. For the critical strongly coupled case, numerical
computations for electromagnetic fields have been presented by Huang et.\ al,
2018. In this paper, we build several theoretical results on the eigenstructure
behavior of the $\gamma$-GEPs. We prove that the $\gamma$-GEP is regular for
any $\gamma > 0$, and the $\gamma$-GEP has $2 \times 2$ Jordan blocks of
infinite eigenvalues at the critical value $\gamma_{*}$. Then, we show that the
$2 \times 2$ Jordan block will split into a complex conjugate eigenvalue pair
that rapidly goes down and up and then collides at some real point near the
origin. Next, it will bifurcate into two real eigenvalues, with one moving
toward the left and the other to the right along the real axis as $\gamma$
increases. A newly formed state whose energy is smaller than the ground state
can be created as $\gamma$ is larger than the critical value. This stunning
feature of the physical phenomenon would be very helpful in practical
applications. Therefore, the purpose of this paper is to clarify the
corresponding theoretical eigenstructure of 3D Maxwell's equations with Pasteur
media.
