Connectivity and Design of Planar Global Attractors of Sturm Type. III: Small and Platonic Examples

Based on a Morse-Smale structure, we study planar global attractors Af of the scalar reaction-advection-diffusion equation ut = uxx + f(x, u, ux) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of f , and hyperbolicity of equilibria. We call Af Sturm attractor because our results strongly rely on nonlinear nodal properties of Sturm type. Planar Sturm attractors Af consist of equilibria, of Morse index 0, 1, or 2, and of the heteroclinic connecting orbits between them. The unique heteroclinic orbits between adjacent Morse levels define a plane graph Cf , which we call the connection graph. The 1-skeleton C f is the closure of the unstable manifolds (separatrices) of the index-1 Morse saddles. We summarize and apply two previous results [FiRo07a, FiRo07b] which completely characterize the connection graphs Cf and their 1-skeletons C f , in purely graph theoretical terms. Connection graphs are characterized by the existence of pairs of Hamiltonian paths with certain chiral restrictions on face passages. Their 1-skeletons are characterized by the existence of cycle-free orientations with certain restrictions on their criticality. We describe all planar Sturm attractors with up to 11 equilibria. We also design planar Sturm attractors with prescribed Platonic 1-skeletons of their connection graphs. We present complete lists for the tetrahedron, octahedron, and cube. We provide representative examples for the design of dodecahedral and icosahedral Sturm attractors.