SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 2030-2058, January 2020.
Spatially localized two-dimensional spot patterns occur for a wide variety of two component reaction-diffusion systems in the singular limit of a large diffusivity ratio. Such localized, far-from-equilibrium patterns are known to exhibit a wide range of different instabilities such as breathing oscillations, spot annihilation, and spot self-replication behavior. Prior numerical simulations of the Schnakenberg and Brusselator systems have suggested that a localized peanut-shaped linear instability of a localized spot is the mechanism initiating a fully nonlinear spot self-replication event. From a development and implementation of a weakly nonlinear theory for shape deformations of a localized spot, it is shown through a normal form amplitude equation that a peanut-shaped linear instability of a steady-state spot solution is always subcritical for both the Schnakenberg and Brusselator reaction-diffusion systems. The weakly nonlinear theory is validated by using the global bifurcation software pde2path [H. Uecker, D. Wetzel, and J. D. Rademacher, Numer. Math. Theory Methods Appl., 7 (2014), pp. 58--106] to numerically compute an unstable, non-radially symmetric, steady-state spot solution branch that originates from a symmetry-breaking bifurcation point.

中文翻译：

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奇摄动反应扩散系统局部斑点的花生形变形的弱非线性分析

SIAM应用动力系统杂志，第19卷第3期，第2030-2058页，2020年1月。
空间分布的二维点状图样以大扩散比的奇异极限出现在各种不同的两组分反应扩散系统中。众所周知，这种局部的，远离平衡的模式会表现出各种不同的不稳定性，例如呼吸振荡，斑点ni灭和斑点自我复制行为。Schnakenberg和Brusselator系统的先前数值模拟表明，局部斑点的局部花生形线性不稳定性是引发完全非线性斑点自我复制事件的机制。从对局部形状变形的弱非线性理论的发展和实施中，通过正态振幅方程可以看出，对于Schnakenberg和Brusselator反应扩散系统，稳态点解的花生形线性不稳定性始终是亚临界的。弱非线性理论通过使用全局分叉软件pde2path验证。Uecker，D.Wetzel和JD Rademacher，Numer。数学。Theory Methods Appl。，7（2014），pp。58--106]，以数值方式计算不稳定的，非径向对称的稳态点解分支，该分支解来自于破坏对称的分支点。