Degenerate bifurcation of the rotating patches

TL;DR

This paper investigates the bifurcation behavior of doubly-connected rotating patches in Euler equations under degenerate conditions, revealing a transcritical bifurcation for two-fold symmetry and non-bifurcation for higher symmetries, aligning with numerical findings.

Contribution

It demonstrates the occurrence of transcritical bifurcation in degenerate cases and clarifies the symmetry-dependent bifurcation behavior, addressing an open problem.

Findings

Transcritical bifurcation for two-fold symmetry doubly-connected patches.

No bifurcation for higher m-fold symmetries with m ≥ 3.

Results align with recent numerical observations.

Abstract

In this paper we study the existence of doubly-connected rotating patches for Euler equations when the classical non-degeneracy conditions are not satisfied. We prove the bifurcation of the V-states with two-fold symmetry, however for higher $m-$fold symmetry with $m\geq3$ the bifurcation does not occur. This answers to a problem left open in \cite{H-F-M-V}. Note that, contrary to the known results for simply-connected and doubly-connected cases where the bifurcation is pitchfork, we show that the degenerate bifurcation is actually transcritical. These results are in agreement with the numerical observations recently discussed in \cite{H-F-M-V}. The proofs stem from the local structure of the quadratic form associated to the reduced bifurcation equation.