Zero-Hopf singularity in bidirectional ring network model with delay

This paper reports a bidirectional ring network model with delay. Zero-Hopf bifurcation is studied by using the center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm forms at the zero-Hopf singularity and show that the model can exhibit pitchfork and Hopf bifurcation. Some numerical simulations are given to support the analytic results, and near the zero-Hopf singularity point, this model displays quasi-periodic, double periodic and multiple periodic trajectory.