Pattern selection at the bifurcation point

Abstract

The bifurcation analysis of inhomogeneous solutions to nonlinear reaction–diffusion equations in the infinite plane is carried out without imposing any a priori symmetry properties. A continuum of solutions bifurcating at the infinitely degenerate eigenvalue is constructed and these solutions are consequently sorted out by the stability analysis. The symmetric honeycomb pattern composed of a triplet of interrelated plane waves emerges as a preferred solution from second-order bifurcation equations, which is further confirmed by the conservation of the hexagonal symmetry in higher orders as well as by the character of fastest growing unstable modes. A condition involving third derivatives of kinetic functions decides whether a ’’crystalline’’ pattern based on a single hexagonal combination or an ’’amorphous’’, or turbulent, composite pattern tends to develop. The transition to an inhomogeneous state, which is normally a first-order, or hard phase transition, becomes a second-order, or soft one at the critical point, where stable small-amplitude patterns bifurcate supercritically under the same condition that is required for the formation of the honeycomb structure beyond the criticality.