  Circular domains frequently appear in the fields of ecology, biology and
chemistry. In this paper, we investigate the equivariant Hopf bifurcation of
partial functional differential equations with Neumann boundary condition on a
two-dimensional disk. The properties of these bifurcations around equilibriums
are analyzed rigorously by studying the equivariant normal forms. Two
reaction-diffusion systems with discrete time delays are selected as numerical
examples to verify the theoretical results, in which spatially inhomogeneous
periodic solutions including standing waves and rotating waves, and spatially
homogeneous periodic solutions are found near the bifurcation points.

Dynamical Systems