  Building on the development of [MOR13], bifurcation of unstable modes that
emerge from continuous spectra in a class of infinite-dimensional noncanonical
Hamiltonian systems is investigated. Of main interest is a bifurcation termed
the continuum Hamiltonian Hopf (CHH) bifurcation, which is an
infinite-dimensional analog of the usual Hamiltonian Hopf (HH) bifurcation.
Necessary notions pertaining to spectra, structural stability, signature of the
continuous spectra, and normal forms are described. The theory developed is
applicable to a wide class of 2+1 noncanonical Hamiltonian matter models, but
the specific example of the Vlasov-Poisson system linearized about homogeneous
(spatially independent) equilibria is treated in detail. For this example,
structural (in)stability is established in an appropriate functional analytic
setting, and two kinds of bifurcations are considered, one at infinite and one
at finite wavenumber. After defining and describing the notion of dynamical
accessibility, Kre\u{i}n-like theorems regarding signature and bifurcation are
proven. In addition, a canonical Hamiltonian example, composed of a negative
energy oscillator coupled to a heat bath, is treated and our development is
compared to pervious work in this context. A careful counting of eigenvalues,
with and without symmetry, is undertaken, leading to the definition of a
degenerate continuum steady state (CSS) bifurcation. It is described how the
CHH and CSS bifurcations can be viewed as linear normal forms associated with
the nonlinear single-wave model described in [BAL13], which is a companion to
the present work and that of [MOR13].
