Bautin bifurcation in a class of two-neuron networks with resonant bilinear terms

This paper addresses the dynamics of a class of two-neuron networks with resonant bilinear terms, which are of the formx˙1=(α1+a)f(x1)+(α2+b)f(x2)+cx1x2,x˙2=(α2-b)f(x1)+(α1-a)f(x2)+dx1x2,where α1 and α2 are two independent parameters. Our main contribution is to present a sufficient condition for the occurrence of a Bautin bifurcation in such a network by using the standard normal form theory and with the Maple software. Two numerical examples are given to demonstrate the utility of our result. To our knowledge, this is the first time to study Bautin bifurcation for recurrent neural networks.