Continuum Hamiltonian Hopf Bifurcation I

TL;DR

This paper explores Hamiltonian bifurcations in noncanonical Hamiltonian matter models, analyzing both discrete and continuous spectra in multi-fluid systems and establishing a foundation for Kren-like theorems related to the Hamiltonian Hopf bifurcation.

Contribution

It introduces a comprehensive framework for understanding Hamiltonian bifurcations with discrete and continuous spectra in multi-fluid models and sets the stage for rigorous Kren-like theorems.

Findings

Identification of Hamiltonian bifurcations in multi-fluid models

Analysis of spectra with steady state and Hamiltonian Hopf bifurcations

Development of signature attachment to continuous spectra

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}.