Singular limit in Hopf bifurcation for doubly diffusive convection equations I: linearized analysis at criticality

TL;DR

This paper analyzes the spectral properties of a singularly perturbed artificial compressible system for doubly diffusive convection near Hopf bifurcation points, laying groundwork for understanding nonlinear bifurcations and singular limits.

Contribution

It provides a detailed spectral analysis of the linearized operator at criticality for a singular perturbation of doubly diffusive convection equations.

Findings

Spectrum characterized near bifurcation point

Foundation for nonlinear bifurcation analysis

Insights into singular limit behavior

Abstract

A singularly perturbed system for doubly diffusive convection equations, called the artificial compressible system, is considered on a two-dimensional infinite layer for a parameters range where the Hopf bifurcation occurs in the corresponding incompressible system. The spectrum of the linearized operator in a time periodic function space is investigated in detail near the bifurcation point when the singular perturbation parameter is small. The results of this paper are the basis of the study of the nonlinear Hopf bifurcation problem and the singular limit of the time periodic bifurcating solutions.