Codimension one and two bifurcations in a symmetrical ring network with delay

In this paper, we discussed all the possible codimension-one and partial codimension-two bifurcations existing in a symmetrical ring network, using the symmetric bifurcation theory of delay differential equations coupled with the representation theory of Lie groups, and dynamical analysis methods. We have not only figured out the pattern of each bifurcation, but also predicted the directions and stability analysis of the bifurcated solutions according to structure of the system. Numerical simulations were then given to verify our theoretical analysis and investigate the complexity of codimension-two mode intersections. Comments were also presented to highlight some future work.