反应-扩散系统中波分岔的振幅方程
Amplitude equations for wave bifurcations in reaction-diffusion systems
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中文总结 AI 辅助
本文推导了反应-扩散系统中波分岔的弱非线性规范形系数至五阶的完整公式，适用于任意n组分系统，并通过软件实现，在多个模型上验证了其与Hopf、图灵分岔的联系。
中文摘要 AI 辅助
波分岔是反应-扩散系统中图灵不稳定性的对应物，但其中临界波数对应于一对纯虚数，而非零时间特征值。此类分岔至少需要三个组分，并产生在空间和时间上均呈周期性的图案。根据边界条件的不同，这些图案可以包含旋转波或驻波。限制在一维空间系统中，我们推导了该分岔的弱非线性规范形系数评估的完整公式，最高阶至五阶，包括决定旋转波和驻波临界性的系数。这些公式适用于任意$n$组分系统（$n\geq 3$），其评估已在软件中实现，并作为补充材料提供。该理论在可激发介质的两个不同版本的三组分反应-扩散模型上进行了说明，这些模型先前已被证明具有超临界和次临界波不稳定性，并在一个五组分的双层化学反应模型上进行了说明。在每种情况下，都生成了双参数分岔图，以说明复杂色散关系与不同类型的Hopf、图灵和波分岔之间的联系，包括若干余维二分岔的存在。
英文摘要
A wave bifurcation is the counterpart to a Turing instability in reaction-diffusion systems, but where the critical wavenumber corresponds to a pure imaginary pair rather than a zero temporal eigenvalue. Such bifurcations require at least three components and give rise to patterns that are periodic in both space and time. Depending on boundary conditions, these patterns can comprise either rotating or standing waves. Restricting to systems in one spatial dimension, complete formulae are derived for the evaluation of the coefficients of the weakly nonlinear normal form of the bifurcation up to order five, including those that determine the criticality of both rotating and standing waves. The formulae apply to arbitrary $n$-component systems ($n\geq 3$) and their evaluation is implemented in software which is made available as supplementary material. The theory is illustrated on two different versions of three-component reaction-diffusion models of excitable media that were previously shown to feature super- and subcritical wave instabilities and on a five-component model of two-layer chemical reaction. In each case, two-parameter bifurcation diagrams are produced to illustrate the connection between complex dispersion relations and different types of Hopf, Turing, and wave bifurcations, including the existence of several codimension-two bifurcations.
发表机构
- University of Bristol（布里斯托大学）
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