Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes   Examples complete our trilogy on the geometric and combinatorial
characterization of global Sturm attractors $\mathcal{A}$ which consist of a
single closed 3-ball. The underlying scalar PDE is parabolic, $$ u_t = u_{xx} +
f(x,u,u_x)\,, $$ on the unit interval $0 < x<1$ with Neumann boundary
conditions. Equilibria $v_t=0$ are assumed to be hyperbolic. Geometrically, we
study the resulting Thom-Smale dynamic complex with cells defined by the fast
unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a
regular cell complex. In the first two papers we characterized 3-ball Sturm
attractors $\mathcal{A}$ as 3-cell templates $\mathcal{C}$. The
characterization involves bipolar orientations and hemisphere decompositions
which are closely related to the geometry of the fast unstable manifolds. An
equivalent combinatorial description was given in terms of the Sturm
permutation, alias the meander properties of the shooting curve for the
equilibrium ODE boundary value problem. It involves the relative positioning of
extreme 2-dimensionally unstable equilibria at the Neumann boundaries $x=0$ and
$x=1$, respectively, and the overlapping reach of polar serpents in the
shooting meander. In the present paper we apply these descriptions to
explicitly enumerate all 3-ball Sturm attractors $\mathcal{A}$ with at most 13
equilibria. We also give complete lists of all possibilities to obtain solid
tetrahedra, cubes, and octahedra as 3-ball Sturm attractors with 15 and 27
equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and
dodecahedra, we indicate a reduction to mere planar considerations as discussed
in our previous trilogy on planar Sturm attractors.

14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
21
The categories of this paper are: adap-org
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph