Pattern formation on regular polygons and circles

Woolley, Thomas ORCID: https://orcid.org/0000-0001-6225-5365 2025. Pattern formation on regular polygons and circles. Journal of Nonlinear Science 35 , 5. 10.1007/s00332-024-10096-6

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Abstract

We investigate the formation of Turing patterns on regular polygonal domains, as the number of edges grow, leading to the limiting case of the circle. Using linear and weakly nonlinear analysis, and evidence by simulations, we demonstrate how the domain shape can fundamentally change the expected bifurcation structure. Specifically, on the square domain we are able to derive pitchfork bifurcations for stripe and spot solutions, as well as show that both branches cannot bifurcate to produce stable patterns. This compares with the case of the equilateral triangle domain that causes the Turing bifurcation to be generically transcritical and, in some cases, none of the bifurcating branches are stable. Moreover, we find a monotonically increasing, but nonlinear relationship, between the minimal bifurcation area and the number of edges. Thus, patterns can occur on triangles with much smaller areas than circles. Overall, this work raises questions for researchers who are simulating applications on domains with simple shapes. Specifically, even small changes to domain geometry can have large impacts on the produced patterns; thus, domain perturbations should be considered in any sensitivity analyses.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Springer Verlag (Germany)
ISSN:	0938-8974
Date of First Compliant Deposit:	18 October 2024
Date of Acceptance:	11 October 2024
Last Modified:	02 Dec 2024 14:43
URI:	https://orca.cardiff.ac.uk/id/eprint/173128

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