Pattern Formation in a Reaction-Diffusion BAM Neural Network With Time Delay: (<i>k</i> <sub>1</sub>, <i>k</i> <sub>2</sub>) Mode Hopf-Zero Bifurcation Case
Tao Dong, Weilai Xiang, Tingwen Huang, Huaqing Li
Abstract
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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$({k_{1}},{k_{2}})$ </tex-math></inline-formula> mode Hopf-zero bifurcation. First, the conditions for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${k_{1}}$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${k_{2}}$ </tex-math></inline-formula> mode Hopf bifurcation and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$({k_{1}},{k_{2}})$ </tex-math></inline-formula> 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.