Stability and Hopf Bifurcation in the General Langford System

This paper is concerned with the general Langford system under homogeneous Neumann boundary conditions. The stabilities of constant solutions are discussed for the general Langford ODE and PDE systems, respectively. Based on the stability results, for the Langford ODE system, the existence, bifurcation direction and stability of periodic solutions are established. Then for the Langford PDE system, by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous periodic solutions are investigated. Finally, numerical simulations are shown to support and supplement the results of theoretical analysis.