奇摄动反应扩散系统花生形变形的弱非线性分析
Weakly Nonlinear Analysis of Peanut-Shaped Deformations for Localized Spots of Singularly Perturbed Reaction-Diffusion Systems
摘要：
空间局部化的2-D斑点图案发生在各种各样的两个组件的反应扩散系统在一个大的扩散率比的奇异极限。这种局部的、远离平衡的图案已知表现出广泛的不同不稳定性，例如呼吸振荡、斑点湮灭和斑点自复制行为。先前的数值模拟的Schnakenberg和Escherselator系统已经表明，一个本地化的花生形的线性不稳定性的本地化点的机制启动一个完全非线性的点自复制事件。从局部化斑点形状变形的弱非线性理论的发展和实现，它是通过一个正常形式的振幅方程，花生形的线性不稳定性的稳态斑点解决方案总是亚临界的Schnakenberg和Schnakelator反应扩散系统。利用全局分岔软件{ \em pde 2 path} [H.~ Uecker等人，数字数学：Theory，Methods and Applications，{ \bf 7}（1），（2014）]，以数值计算源自于非对称性断裂分叉点的不稳定、非径向对称、稳态点解分支。
英文摘要：
Spatially localized 2-D 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 {\em pde2path} [H.~Uecker et al., Numerical Mathematics: Theory, Methods and Applications, {\bf 7}(1), (2014)] to numerically compute an unstable, non-radially symmetric, steady-state spot solution branch that originates from a symmetry-breaking bifurcation point.
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