Pattern Formation in a Reaction-Diffusion BAM Neural Network With Time Delay: (k1, k2) Mode Hopf-Zero Bifurcation Case

This article investigates the joint effects of connection weight and time delay on pattern formation for a delayed reaction-diffusion BAM neural network (RDBAMNN) with Neumann boundary conditions by using the ({k_{1}},{k_{2}}) mode Hopf-zero bifurcation. First, the conditions for {k_{1}} mode zero bifurcation are obtained by choosing connection weight as the bifurcation parameter. It is found that the connection weight has a great impact on the properties of steady state. With connection weight increasing, the homogeneous steady state becomes inhomogeneous, which means that the connection weight can affect the spatial stability of steady state. Then, the specified conditions for the {k_{2}} mode Hopf bifurcation and the ({k_{1}},{k_{2}}) mode Hopf-zero bifurcation are established. By using the center manifold, the third-order normal form of the Hopf-zero bifurcation is obtained. Through the analysis of the normal form, the bifurcation diagrams on two parameters' planes (connection weight and time delay) are obtained, which contains six areas. Some interesting spatial patterns are found in these areas: a homogeneous periodic solution, a homogeneous steady state, two inhomogeneous steady state, and two inhomogeneous periodic solutions.