Nonlinear waves in complex oscillator network with delay

•Calculate the critical values of Hopf bifurcation.•Prove that the existence of multiple periodic solutions.•The stability of these nonlinear oscillations is determined.

We investigate the spatio-temporal patterns of Hopf bifurcating periodic solutions in a delay complex oscillator network. Firstly, we calculate the critical values of Hopf bifurcation. Secondly, the bifurcating periodic solutions can take on two cases: one is synchronization or anti-synchronization, and another is the coexistence of two phase-locked, N mirror-reflecting and N standing waves, because the system has group symmetry. Finally, the stability of these nonlinear oscillations is determined using the center manifold theorem and normal form method with the imaginary eigenvalues being simple and double.