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<p>This paper considers a diffusive Brusselator system subject to homogeneous Neumann boundary condition on a planar rectangle domain. The spatially homogeneous Hopf bifurcation curve and Turing bifurcation curve of system at the unique positive constant steady-state are derived by analyzing in detail the distribution in the complex plane of roots of the associated eigenvalue problem. When the concentrations of two input reactants are restricted interior the Turing unstable domain and near the Turing bifurcation curve, by choosing the ratio of the diffusion coefficients of two reactants as the bifurcation parameter, the cubic normal form of Turing bifurcation of system at the positive constant steady-state is established according to the normal form method and the center manifold theorem for reaction-diffusion systems subject to homogeneous Neumann boundary condition. Based on the obtained normal form, we find that under suitable conditions, the diffusive Brusselator system can bifurcate two symmetric spatially heterogeneous steady-states from the positive constant one and numerical simulations used to verify this finding are also carried out. In addition, Turing hexagonal patterns of system near the positive constant steady-state are explored. Firstly, the amplitude equations of Turing hexagonal patterns of system near the positive constant steady-state are derived on the basis of the multiple-scale perturbation analysis combined with the successive approximation method. Then by analyzing the existence and stability of the stationary solutions of the amplitude equations, the existence and stability of typical Turing hexagonal patterns such as hexagons, stripes and the mixture of hexagons and stripes are demonstrated. The theoretical results show that spatial diffusion in a reaction-diffusion system can generate more complicated dynamical behaviors and also provide a theoretical basis for the mechanism of formation of spatial structures in nonlinear systems. To verify the validity of our theoretical predictions, numerical simulations are also provided with the help of the finite difference method solving the numerical solutions of parabolic partial differential equations.</p>

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Turing hexagonal patterns in a diffusive Brusselator system with Neumann boundary conditions on a plane domain

  • 期刊论文

  • 发表日期：

    2026年4月2日

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  • Xiang-Ping Yan,
  • Hao-Hao Qian,
  • Cun-Hua Zhang

摘要

This paper considers a diffusive Brusselator system subject to homogeneous Neumann boundary condition on a planar rectangle domain. The spatially homogeneous Hopf bifurcation curve and Turing bifurcation curve of system at the unique positive constant steady-state are derived by analyzing in detail the distribution in the complex plane of roots of the associated eigenvalue problem. When the concentrations of two input reactants are restricted interior the Turing unstable domain and near the Turing bifurcation curve, by choosing the ratio of the diffusion coefficients of two reactants as the bifurcation parameter, the cubic normal form of Turing bifurcation of system at the positive constant steady-state is established according to the normal form method and the center manifold theorem for reaction-diffusion systems subject to homogeneous Neumann boundary condition. Based on the obtained normal form, we find that under suitable conditions, the diffusive Brusselator system can bifurcate two symmetric spatially heterogeneous steady-states from the positive constant one and numerical simulations used to verify this finding are also carried out. In addition, Turing hexagonal patterns of system near the positive constant steady-state are explored. Firstly, the amplitude equations of Turing hexagonal patterns of system near the positive constant steady-state are derived on the basis of the multiple-scale perturbation analysis combined with the successive approximation method. Then by analyzing the existence and stability of the stationary solutions of the amplitude equations, the existence and stability of typical Turing hexagonal patterns such as hexagons, stripes and the mixture of hexagons and stripes are demonstrated. The theoretical results show that spatial diffusion in a reaction-diffusion system can generate more complicated dynamical behaviors and also provide a theoretical basis for the mechanism of formation of spatial structures in nonlinear systems. To verify the validity of our theoretical predictions, numerical simulations are also provided with the help of the finite difference method solving the numerical solutions of parabolic partial differential equations.