Chaotic Turing patterns in the Brusselator on axisymmetric surfaces

Determine whether the Brusselator reaction–diffusion system on axisymmetric curved surfaces exhibits chaotic spatiotemporal Turing patterns corresponding to the chaotic solutions of the two-mode amplitude equations derived for axisymmetric geometries, and, if so, identify the surface geometries and parameter regimes that realize such chaos in the full Brusselator model.

Background

The paper derives two-mode amplitude equations to describe weakly nonlinear Turing pattern dynamics on axisymmetric surfaces and shows that these reduced equations can support complex behaviors, including limit cycles and chaos, depending on surface-induced coupling terms and coefficients.

While limit-cycle (oscillatory) propagation predicted by the amplitude equations is validated by direct simulations of the Brusselator model on axisymmetric surfaces, the authors report that they have not yet realized chaotic dynamics in the full Brusselator system under axisymmetry. They do observe chaotic behavior on a non-axisymmetric, deformed sphere, indicating that geometry can induce chaos, but whether such chaos occurs for axisymmetric surfaces in the full model remains unresolved.