Computation of Turing Bifurcation Normal Form for n-Component Reaction-Diffusion Systems

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 codimensio...

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Bibliographic Details
Published in:ACM transactions on mathematical software 2023-12, Vol.49 (4), p.1-24, Article 35
Main Authors: Villar-Sepúlveda, Edgardo, Champneys, Alan
Format: Article
Language:English
Subjects:
Applied computing
Mathematics and statistics
Mathematics of computing
Physics
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Summary: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.
ISSN:0098-3500
1557-7295
DOI:10.1145/3625560