The Hopf Bifurcation In A Planar Differential System With Delay

In this paper, we consider the following planar systemwhere x(t) = (x1(t),x2(t))T R2, t > 0 is a real constant, F:R2xR2 -R2 and the origin is an isolated equilibrium. Throughout this paper, we assume that the function F satisfies some conditions under which the linearized equations of (1) at the origin iswhere ai,bi,ci,di, (i = 1,2) are real constants. The planar delay differential system (1) has significant biological and physical backgrounds. For example, some special cases of (1) have been proposed as models of neural networks. In this thesis, we take the delay r as a bifurcation parameter to investigate the Hopf bifurcation phenomenon in system (1). By analyzing the associated characteristic transcendental equation of system (2), and using the Hopf bifurcation theorem, we obtain one condition for the existence of Hopf bifurcation in system (1). Furthermore, based on the center manifold theorem and the method of normal form, some interesting results about the properties of Hopf bifurcation are obtained, including the direction of Hopf bifurcation and stability of Hopf bifurcating periodic solutions. At last, we give a Cohen-Grossberg neural network model as a specific example to illustrate the obtained results.It should be mentioned that, by making some transformations, some planar differential systems arising form neural network models with two different delays can be written as system (1). In addition, one of the main differences between our study and the existing research on system (1) is that we don't require the parameters in the linearized equations (2) of system (1) at the origin to satisfy b1 = a2 =0, a1 = b2 < 0.