General expressions are derived for the amplitude equation valid at a Turing bifurcation of a system of reaction-diffusion equations in one spatial dimension, with an arbitrary number of components. The normal form is computed up to fifth order, which enables the detection and analysis of codimension-two points where the criticality of the bifurcation changes. The expressions are implemented within a Python package, in which the user needs to specify only expressions for the reaction kinetics and the values of diffusion constants. The code is augmented with a Mathematica routine to compute curves of Turing bifurcations in a parameter plane and automatically detect codimension-two points. The software is illustrated with examples that show the versatility of the method including a case with cross-diffusion, a higher-order scalar equation and a four-component system.

推导了在一个空间维度上具有任意数量的分量的反应扩散方程组的图灵分岔处有效的振幅方程的一般表达式。范式的计算最高可达五阶，这使得能够检测和分析余维（分岔临界性发生变化的两点）。这些表达式在 Python 包中实现，用户只需指定反应动力学表达式和扩散常数值。该代码通过 Mathematica 例程进行了增强，可计算参数平面中的图灵分岔曲线并自动检测余维两点。该软件通过示例进行了说明，显示了该方法的多功能性，包括交叉扩散的情况、高阶标量方程和四分量系统。