Bogdanov–Takens bifurcation in a tri-neuron BAM neural network model with multiple delays

In this paper, a tri-neuron BAM neural network model with multiple delays is considered. We show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. There is a wide range of different dynamical behaviors which can be produced by varying the coupling strength. By choosing the connected weights c ₂₁ and c ₃₁ (the connection weights through the neurons from J-layer to I-layer) as bifurcation parameters, the critical values where a Bogdanov–Takens bifurcation occurs are derived. Then, by computing the normal forms for the system, the bifurcation diagrams are obtained. Furthermore, some interesting phenomena, such as saddle-node bifurcation, pitchfork bifurcation, homoclinic bifurcation, heteroclinic bifurcation and double limit cycle bifurcation are found by choosing the different connection strengths. Some numerical simulations are given to support the analytic results.