Approximate localised dihedral patterns near a turing instability

Abstract Fully localised patterns involving cellular hexagons or squares have been found experimentally and numerically in various continuum models. However, there is currently no mathematical theory for the emergence of these from a quiescent state. A key issue that standard techniques one-dimensional proven insufficient understanding localisation higher dimensions. In this work, we present comprehensive approach to problem by using developed study radially-symmetric patterns. Our analysis covers planar equipped with wide range dihedral symmetries, thereby avoiding restriction solutions on predetermined lattice. The context paper such near Turing instability general class reaction-diffusion equations. Posing system polar coordinates, carry out finite-mode Fourier decomposition angular variable yield large coupled radial ordinary differential We then utilise spatial dynamics methods, as invariant manifolds, rescaling charts, normal form analysis, leading an algebraic matching condition exist reduction. This nontrivial, which solve via combination by-hand calculations Gröbner bases polynomial algebra reveal existence plethora These results capture essence emergent hexagonal witnessed experiments. Moreover, combine computer-assisted Newton–Kantorovich procedure prove patches 6 m -fold symmetry arbitrarily decompositions. includes hexagon elusive analytical treatment.