An Efficient Method for Studying fold-Hopf bifurcation in Delayed Neural Networks

An effective and simple method, called perturbation scheme (PS), is proposed to study fold-Hopf bifurcation for delayed neural systems qualitatively and quantitatively when time delay and connection weight are considered as two bifurcation parameters. As an illustration, the proposed method is employed to investigate a delayed bidirectional associative memory (BAM) neural network. Dynamics arising from fold-Hopf bifurcation are classified qualitatively and expressed approximately in a closed form for periodic solution. We also give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the delayed system. Our investigations reveal that secondary Hopf bifurcation and pitchfork bifurcation of limit cycle may emanate from the pitchfork-Hopf point. In addition, the secondary Hopf bifurcation can lead to multistability between equilibrium points and periodic solution in some region of parameter space. The validity of analytical predictions is shown by their consistency with the results from the center manifold reduction (CMR) with normal form and numerical simulation. As an analytical tool, the advantage of the PS also lies in its simplicity and ease of implementation.