A Degenerate Hopf Bifurcation Theorem in Infinite Dimensions

A Hopf bifurcation theorem is established for the abstract evolution equation $\frac{\mathrm{d}x}{\mathrm{d}t}=F(x,λ)$ in infinite dimensions under the degeneracy condition $Re μ^{\prime}(λ_0)= 0$ 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 $λ_0$ although the stability property of the trivial solutions does not change near $λ_0$. 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.

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