  We demonstrate several new aspects of exceptional points of degeneracy (EPD)
pertaining to propagation in two uniform coupled transmission-line structures.
We describe an EPD using two different approaches - by solving an eigenvalue
problem based on the system matrix, and as a singular point from bifurcation
theory, and the link between these two disparate viewpoints. Cast as an
eigenvalue problem, we show that eigenvalue degeneracies are always coincident
with eigenvector degeneracies, so that all eigenvalue degeneracies are
implicitly EPDs in two uniform coupled transmission lines. Furthermore, we
discuss in some detail the fact that EPDs define branch points (BPs) in the
complex-frequency plane; we provide simple formulas for these points, and show
that parity-time (PT) symmetry leads to real-valued EPDs occurring on the
real-frequency axis. We discuss the connection of the linear algebra approach
to previous waveguide analysis based on singular points from bifurcation
theory, which provides a complementary viewpoint of EPD phenomena, showing that
EPDs are singular points of the dispersion function associated with the fold
bifurcation. This provides an important connection of various modal interaction
phenomena known in guided-wave structures with recent interesting effects
observed in quantum mechanics, photonics, and metamaterials systems described
in terms of the EPD formalism.
