Pitchfork and Hopf bifurcation in a delayed neural network system

In this paper, a neural network system with the different delay-couplings has been established. The effects of the time delays on system dynamics are investigated. The multiple delays greatly complicate the analysis of stability and Hopf bifurcation. By analyzing the corresponding characteristic equation of the trivial equilibrium point, the delay-independent instability criteria are derived, i.e. under which the system will remain the instability for arbitrary delays. Then the conditions of delay-dependent asymptotic stability are investigated in detail. The periodical behaviors can be found when time delays are fixed in the unstable regions. Furthermore, we find that the system exhibits the multiple switches of the stability and instability. At last, some numerical simulations are included to illustrate our theory results.