退化图灵分岔与激活-抑制系统中局域斑图的诞生
Degenerate Turing bifurcation and the birth of localised patterns in activator-inhibitor systems
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中文总结 AI 辅助
本文通过两种参数标度下的规范形计算，证明了激活-抑制系统中图灵分岔的亚临界与超临界性质，并揭示余维二退化分岔及局域斑图的诞生机制。
中文摘要 AI 辅助
本文为一维无限域上一类一般的激活-抑制反应扩散方程中图灵分岔的存在性和临界性提供了精确条件。该类方程包括广义的Schnakenberg和Brusselator模型，以及具有三次自催化非线性项的其他模型。先前的数值工作表明，由于所谓的同宿蜿蜒机制，存在包含局域斑图的分岔结构。本文提供了显式计算来证明这些结果。在扩散比$\delta \to 0$的极限下，考虑两种不同的参数标度，它们导致可处理的规范形系数。首先，在小参数标度下，图灵分岔被证明总是亚临界的。其次，大参数标度揭示图灵分岔是超临界的，通过连续性，导致余维二退化分岔的存在。还计算了一个5阶规范形系数的符号，该符号被证明对于同宿蜿蜒的局域诞生具有正确的符号。对于Brusselator的情况，可以显式地找到两个这样的余维二点，以及Maxwell点的首阶表达式，在该参数楔形区域中局域斑图出现。数值结果与理论一致。
英文摘要
Precise conditions are provided for the existence and criticality of Turing bifurcations in a general class of activator-inhibitor reaction-diffusion equations on a one-dimensional infinite domain. The class includes generalised Schnakenberg and Brusselator models, as well as other models with cubic autocatalytic nonlinear terms. Previous numerical work suggests the existence of a bifurcation structure containing localised patterns due to the so-called homoclinic snaking mechanism. This paper provides explicit calculations to justify those results. Two distinct scalings of parameters that lead to tractable normal-form coefficients are considered in the limit that the diffusion ratio $δ\to 0$. First, under a small-parameter scaling, the Turing bifurcation is shown to be always subcritical. Second, a large-parameter scaling reveals the Turing bifurcation to be supercritical, leading, by continuity, to the existence of a codimension-two degenerate bifurcation. The sign of a 5th-order normal form coefficient is also computed, which is shown to have the correct sign for the local birth of homoclinic snaking. For the case of the Brusselator, two such codimension-two points can be found explicitly, as can the leading-order expression for the Maxwell point, in a parameter wedge about which localised patterns emerge. Numerical results are found to be consistent with the theory.
发表机构
- University of Bristol（布里斯托大学）
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