Multiple bifurcation analysis in a ring of delay coupled oscillators with neutral feedback

Spatiotemporal periodic patterns, including phase-locked oscillations, mirror-reflecting waves, standing waves, in-phase or anti-phase oscillations are investigated in a ring of bidirectionally coupled oscillators with neutral delay feedback. It is confirmed that neutral feedback makes Hopf bifurcation occur in a larger domain of parameters. We calculate the normal forms near Hopf bifurcation, D N equivariant Hopf bifurcation and double-Hopf bifurcation in this neutral equation by using the method of multiple scales. Theoretically, the appearance of the in-phase, anti-phase and phase-locked oscillations that we observed in the simulation about a ring of delay coupled Hindmarsh–Rose neurons with neutral feedback is explained.