Pattern Formation on Regular Polygons and Circles

The study explores Turing pattern formation on regular polygonal domains, transitioning to circles as the number of edges increases. Through linear and weakly nonlinear analysis, along with simulations, it is shown that domain shape significantly affects bifurcation structures. On square domains, pitchfork bifurcations for stripe and spot solutions are derived, but neither can produce stable patterns. In contrast, equilateral triangle domains lead to generically transcritical Turing bifurcations, often resulting in unstable branches. A nonlinear relationship is found between minimal bifurcation area and edge number, allowing patterns on smaller triangular areas compared to circles. The research highlights the importance of considering domain geometry changes in simulations, as they can greatly influence pattern outcomes.