Continuum Hamiltonian Hopf Bifurcation I

Abstract

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}.