Bifurcation and control in a neural network with small and large delays

This paper investigates a neural network modeled by a scalar delay differential equation. The focus is placed upon the Hopf bifurcation generated by varying the interaction parameter. A general expression for the periodic solutions arising from the Hopf bifurcation is obtained, and the direction of the bifurcation is also determined. Then, our results are tested in the two limits of small and large delays. For small delays, it is shown that a Hopf bifurcation to sinusoidal oscillations emerges as long as the interaction parameter is large enough (bifurcation from infinity) (Rosenblat & Davis, 1979). For large delays, it is pointed out that the oscillation progressively changes from sine to square-wave (Chow, Hale, & Huang, 1992; Hale & Huang, 1994). Moreover, a time delayed feedback control algorithm is introduced to generate the Hopf bifurcation at a desired bifurcation point for our neural network model. It is shown that the linear gain regulates the onset of the bifurcation, while the nonlinear gains govern the direction and the stability of the periodic solutions generated from the Hopf bifurcation.