Bifurcations of a fractional three-layer neural network with different delays: Delay-dependent and order-dependent

In this paper, a novel fractional three-layer delayed neural network is proposed, which is a generalization of some existing BP neural networks and has greater information acquiring ability and fault tolerance. First, the sum of the involved transmission delay and feedback delay is taken as the bifurcation parameter and some delay-induced Hopf bifurcation criteria is given based on the stability theorem of linear system and Hopf bifurcation theorem. Then, as an extension of the traditional delay-induced Hopf bifurcation, the order-induced Hopf bifurcation is further explored by combining implicit function array solution method. Finally, the application and validity of our presented theoretical results are illustrated by two numerical examples. Moreover, the impact of the time delays on the order-induced Hopf bifurcation is deeply discussed through computation and illustration. It is discovered that the critical value of the fractional order becomes smaller when the sum of the involved transmission delay and feedback delay increases.