A Degenerate Hopf Bifurcation Theorem in Infinite Dimensions

TL;DR

This paper extends Hopf bifurcation theory to infinite-dimensional systems under degeneracy conditions, revealing new bifurcation phenomena and applying results to predator-prey models.

Contribution

It establishes a Hopf bifurcation theorem in infinite dimensions with degeneracy, without requiring analyticity, and explores stability and new bifurcation branches.

Findings

Degenerate Hopf bifurcation theorem in infinite dimensions.

Existence of transcritical Hopf bifurcation despite unchanged stability.

Discovery of new periodic solution branches in predator-prey systems.

Abstract

A Hopf bifurcation theorem is established for the abstract evolution equation $\frac{\mathrm{d}x}{\mathrm{d}t}=F(x,\lambda)$ in infinite dimensions under the degeneracy condition $Re \mu ^{\prime}(\lambda_0)= 0$ and suitable assumptions. The stability properties of bifurcating periodic solutions are also derived. Interestingly, it is shown that a transcritical Hopf bifurcation still can occur at $\lambda_0$ although the stability property of the trivial solutions does not change near $\lambda_0$. Our results do not require the analyticity of $F$. The main tools are the Lyapunov--Schmidt reduction and a Morse lemma. Applications to a multi-parameter diffusive predator--prey system discover new branches of periodic solutions.