Sturm 3-ball global attractors 3: Examples

1 Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors A which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, ut = uxx + f(x, u, ux) , on the unit interval 0 < x < 1 with Neumann boundary conditions. Equilibria 2 vt = 0 are assumed to be hyperbolic. 3 4 Geometrically, we study the resulting Thom-Smale dynamic complex with cells 5 defined by the fast unstable manifolds of the equilibria. The Thom-Smale com6 plex turns out to be a regular cell complex. In the first two papers we char7 acterized 3-ball Sturm attractors A as 3-cell templates C. The characterization 8 involves bipolar orientations and hemisphere decompositions which are closely 9 related to the geometry of the fast unstable manifolds. 10 11 An equivalent combinatorial description was given in terms of the Sturm per12 mutation, alias the meander properties of the shooting curve for the equilibrium 13 ODE boundary value problem. It involves the relative positioning of extreme 214 dimensionally unstable equilibria at the Neumann boundaries x = 0 and x = 1, 15 respectively, and the overlapping reach of polar serpents in the shooting meander. 16 17 In the present paper we apply these descriptions to explicitly enumerate all 3-ball 18 Sturm attractors A with at most 13 equilibria. We also give complete lists of 19 all possibilities to obtain solid tetrahedra, cubes, and octahedra as 3-ball Sturm 20 attractors with 15 and 27 equilibria, respectively. For the remaining Platonic 21 3-balls, icosahedra and dodecahedra, we indicate a reduction to mere planar 22 considerations as discussed in our previous trilogy on planar Sturm attractors. 23