PATTERN FORMATION IN A RING NETWORK WITH DELAY

Abstract

We consider a ring network of three identical neurons with delayed feedback. Regarding the coupling coefficients as bifurcation parameters, we obtain codimension one bifurcation (including a Fold bifurcation and Hopf bifurcation) and codimension two bifurcations (including Fold–Fold bifurcations, Fold–Hopf bifurcations and Hopf–Hopf bifurcations). We also give concrete formulas for the normal form coefficients derived via the center manifold reduction that provide detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstable equilibria, periodic solutions, and quasi-periodic solutions.