Singular limit in Hopf bifurcation for doubly diffusive convection equations II: bifurcation and stability

TL;DR

This paper investigates how Hopf bifurcation behavior in doubly diffusive convection systems persists and converges in the singular limit from artificial compressible to incompressible models, revealing stability and bifurcation properties.

Contribution

It demonstrates that Hopf bifurcation and associated periodic solutions in the artificial compressible system converge to those in the incompressible system as the artificial Mach number approaches zero.

Findings

Hopf bifurcation occurs in both systems for small Mach number

Time periodic solutions in the compressible system converge to incompressible bifurcations

Bifurcation structure is preserved in the singular limit

Abstract

A singular perturbation problem from the artificial compressible system to the incompressible system is considered for a doubly diffusive convection when a Hopf bifurcation from the motionless state occurs in the incompressible system. It is proved that the Hopf bifurcation also occurs in the artificial compressible system for small singular perturbation parameter, called the artificial Mach number. The time periodic solution branch of the artificial compressible system is shown to converge to the corresponding bifurcating branch of the incompressible system in the singular limit of vanishing artificial Mach number.