Continuum Hamiltonian Hopf Bifurcation II

TL;DR

This paper develops a theory for continuum Hamiltonian Hopf bifurcations in infinite-dimensional Hamiltonian systems, applying it to the Vlasov-Poisson system and comparing with canonical examples.

Contribution

It introduces a comprehensive framework for analyzing CHH bifurcations, including spectral, stability, and normal form analysis, extending previous finite-dimensional results to infinite-dimensional systems.

Findings

Established structural stability criteria for the Vlasov-Poisson system.

Proved Kren-like theorems relating signature and bifurcation.

Identified and characterized degenerate CSS bifurcations.

Abstract

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].