2022/03/31 by Hongjing Pan, Pan, Hongjing, Ruixiang Xing +3
Medicine · Biochemistry, Genetics and Molecular Biology · Mathematics · #Mathematical and Theoretical Epidemiology and Ecology Models #Evolution and Genetic Dynamics #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.2203.16878
A Hopf bifurcation theorem is established for the abstract evolution equation (dx)/(dt)=F(x,λ) in infinite dimensions under the degeneracy condition Re μ′(λ0)= 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 λ0 although the stability property of the trivial solutions does not change near λ0. 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.