  Hamiltonian bifurcations in the context of noncanonical Hamiltonian matter
models are described. First, a large class of 1 + 1 Hamiltonian multi-fluid
models is considered. These models have linear dynamics with discrete spectra,
when linearized about homogeneous equilibria, and these spectra have
counterparts to the steady state and Hamiltonian Hopf bifurcations when
equilibrium parameters are varied. Examples of fluid sound waves and plasma and
gravitational streaming are treated in detail. Next, using these 1 + 1 examples
as a guide, a large class of 2 + 1 Hamiltonian systems is introduced, and
Hamiltonian bifurcations with continuous spectra are examined. It is shown how
to attach a signature to such continuous spectra, which facilitates the
description of the continuous Hamiltonian Hopf bifurcation. This chapter lays
the groundwork for Kre\u{i}n-like theorems associated with the CHH bifurcation
that are more rigorously discussed in our companion chapter \cite{chaptII}.
