We construct square and target patterns solutions of the FitzHugh–Nagumo reaction–diffusion system on planar bounded domains. We study the existence and stability of stationary square and super-square patterns by performing a close to equilibrium asymptotic weakly nonlinear expansion: the emergence of these patterns is shown to occur when the bifurcation takes place through a multiplicity-two eigenvalue without resonance. The system is also shown to support the formation of axisymmetric target patterns whose amplitude equation is derived close to the bifurcation threshold. We present several numerical simulations validating the theoretical results.

中文翻译：

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二维FitzHugh–Nagumo模型中的模式选择

我们在平面有界域上构造FitzHugh-Nagumo反应扩散系统的正方形和目标图案解。我们通过执行接近平衡渐近的弱非线性扩展来研究固定正方形和超正方形模式的存在性和稳定性：当通过分形二本征值发生分叉而没有共振时，这些模式的出现就发生了。该系统还显示出支持轴对称目标图案的形成，该轴对称目标图案的振幅方程式接近于分叉阈值。我们提出了几个数值模拟，验证了理论结果。