The Hopf bifurcation for the Brusselator ordinary-differential-equation (ODE) model and the corresponding partial-differential-equation (PDE)
model are investigated by using the Hopf bifurcation theorem. The stability of the Hopf bifurcation periodic solution is discussed by applying the normal form theory and the center manifold theorem. When parameters satisfy some
conditions, the spatial homogenous equilibrium solution and the
spatial homogenous periodic solution become unstable. Our results
show that if parameters are properly chosen, Hopf bifurcation does
not occur for the ODE system, but occurs for the PDE system.