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

Abstract

Based on a Morse-Smale structure, we study planar global attractors $${{\mathcal A}_f}$$ of the scalar reaction-advection-diffusion equation u t = u xx + f (x, u, u x ) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of f, and hyperbolicity of equilibria. We call $${{\mathcal A}_f}$$ Sturm attractor because our results strongly rely on nonlinear nodal properties of Sturm type. The planar Sturm attractor $${{\mathcal A}_f}$$ consists of equilibria of Morse index 0, 1, or 2, and their heteroclinic connecting orbits. The unique heteroclinic orbits between adjacent Morse levels define a plane graph $${{\mathcal C}_f}$$ , which we call the connection graph. Its 1-skeleton $${{\mathcal C}^1_f}$$ is the closure of the unstable manifolds (separatrices) of the index-1 Morse saddles. We summarize and apply two previous results (Fiedler and Rocha, J. Diff. Equ. 244: 1255–1286, 2008, Crelle J. Reine Angew. Math. 26 pp., 2009, doi: 10.1515/CRELLE.2009.076 ) which completely characterize the connection graphs $${{\mathcal C}_f}$$ and their 1-skeletons $${{\mathcal C}^1_f}$$ , 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. Such orientations are called bipolar in de Fraysseix et al. (Discrete Appl. Math. 56: 157–179, 1995). 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. Unlike previous examples, and in particular unlike the classification of Sturm attractors with up to nine equilibria, our present results are based on analytic insight rather than mindless computer-based enumeration.