Local bifurcation analysis and global numerical pathfollowing for Turing patterns in 3D reaction--diffusion systems

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Turing patterns are solutions of partial differential equations (PDE) that arise from an instability of a spatially homogeneous stationary solution, which is stable with respect to spatially homogeneous perturbations, but unstable with respect to spatially periodic perturbations. The original modeling was motivated by pattern formation in embryos. However, Turing patterns occur in a variety of systems in nature, and thus also in a variety of PDE models. The local theory is well developed in one or two spatial dimensions, and Turing patterns can be well predicted using amplitude equations near bifurcation from a homogeneous solution. However, many physically relevant systems are genuinely three dimensional (3D), and in 3D the theory becomes much more complicated and is much less developed. Moreover, also numerical calculations of Turing patterns in 3D are rather rare and not systematic. The goal of this project is to use a combination of analysis and numerics to develop tools which allow systematically to study the bifurcation scenario for 3D Turing patterns. Besides the local theory near primary bifurcations, we also aim at a more global picture of the solution space. For this, preparatory work extending the 2D software package pde2path to 3D shall be continued, to also study branches of 3D Turing patterns further away from their primary bifurcation, and to study their secondary and higher order bifurcations, including hetero--and homoclinic connections between different patterns.