Existence of small loops in the Bifurcation diagram near the degenerate eigenvalues

TL;DR

This paper investigates the bifurcation structure of rotating doubly connected patches in incompressible Euler equations, demonstrating that branches with the same symmetry merge into small loops near degenerate eigenvalues, confirming previous numerical findings.

Contribution

It provides a rigorous analysis of the bifurcation diagram near degenerate eigenvalues, showing the formation of small loops in the structure.

Findings

Branches with the same symmetry merge into small loops near degeneracy

The results confirm previous numerical observations

The study advances understanding of bifurcation structures in fluid dynamics

Abstract

In this paper we study for the incompressible Euler equations the global structure of the bifurcation diagram for the rotating doubly connected patches near the degenerate case. We show that the branches with the same symmetry merge forming a small loop provided that they are close enough. This confirms the numerical observations done in the recent work [10]