Hopf bifurcation existence in a delayed neural equation control system

The characteristic equation of a discrete time-delayed continuous system contains the terms of an exponential function with respect to eigenvalues, i.e. the characteristic equation is a transcendental equation. That makes it hard to obtain the conditions for Hopf bifurcation to occur. A system for controlling a neural equation with discrete time delay based on the washout filter is presented. The transcendental equation of the corresponding linearized system is analyzed in detail. In this control system, it is found that a chaotic orbit can be controlled to a periodic solution; Hopf bifurcation occurs when control parameters are properly chosen. Some numerical results are also presented to illustrate the correctness of our results.