Multi-delay-induced bifurcation singularity in two-neuron neural models with multiple time delays

In this article, we mainly concentrate on the impacts of multiple time delays on bifurcation dynamics in two-neuron neural networks including pitchfork bifurcation, Hopf bifurcation and pitchfork-Hopf bifurcation. Specifically, pitchfork-bifurcation close to the trivial equilibrium point is firstly studied by one equilibrium and three equilibrium points. Periodic oscillation and pitchfork-Hopf singularity dependent of time delays are exhibited. Further, on zero-Hopf bifurcation singularity, a methodology-based perturbation is described for the first time to tackle this bifurcation and give some general formulae relating to bifurcation classification and stable/unstable periodic solutions of a high dimension system with multiple time delays. Two-neuron neural system can display complex and rich dynamics in the neighbor of pitchfork-Hopf bifurcation, for example, the coexistence of periodic oscillation and two nontrivial equilibria, multi-stability of two nontrivial equilibria, and stability switching. Theoretical results are in very nice agreement with numerical simulations. The presented methodology avoids the tedious computation from the center manifold reduction and is easier to program to derive delay-induced periodic solutions.