In this talk, after introducing some background about dynamical system,
neural network and delay differential equation, we consider a ring
neural network of identical elements with time delayed, nearest
neighbor coupling. We give some global and local stability results
and show how the presence of time delay allows for mode interactions
leading to the coexistence of different oscillation patterns.
The Hopf and equivariant Hopf bifurcations are analyzed. Regarding
the coupling strengths as bifurcation parameters, we obtain codimension
one bifurcation and the interaction of each critical bifurcations.
Concrete formulae for the normal form coefficients are derived via
the center manifold reduction that provide detailed information about
the bifurcation and stability of various bifurcated solutions.