Abstract
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, \begin{document}$u_t = u_{xx} + f(x, u, u_x)\, , $ \end{document} on the unit interval \begin{document}$0 with Neumann boundary conditions. Equilibria \begin{document}$v_t = 0$\end{document} are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the fast unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. In the first two papers we characterized 3-ball Sturm attractors \begin{document}$\mathcal{A}$\end{document} as 3-cell templates \begin{document}$\mathcal{C}$\end{document} . The characterization involves bipolar orientations and hemisphere decompositions which are closely related to the geometry of the fast unstable manifolds. An equivalent combinatorial description was given in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries \begin{document}$x = 0$\end{document} and \begin{document}$x = 1$\end{document} , respectively, and the overlapping reach of polar serpents in the shooting meander. In the present paper we apply these descriptions to explicitly enumerate all 3-ball Sturm attractors \begin{document}$\mathcal{A}$\end{document} 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.
Highlights
For our general introduction we first follow [FiRo16, FiRo17] and the references there
Sturm global attractors Af are the global attractors of scalar parabolic equations ut = uxx + f (x, u, ux) on the unit interval 0 < x < 1
In [FiRo16, theorem 4.1] we proved that the dynamic complex C:= Cf of a Sturm 3ball Af satisfies properties (i)–(iv) of definition 1.1 on a 3-cell template
Summary
For our general introduction we first follow [FiRo16, FiRo17] and the references there. A finite regular cell complex C coincides with the Thom-Smale dynamic complex cv = W u(v) ∈ Cf of a 3-ball Sturm attractor Af if, and only if, C is a 3-cell template, with the above translation of the hemisphere decomposition of ∂W u(O). In [FiRo17, theorem 2.6], we showed that the Thom-Smale dynamic Sturm complex Cf of Af coincides with the prescribed 3-cell template C, i.e. Cf = C, by a cell homeomorphism which, in addition, preserves the signed hemisphere translation table (1.21). 1.1 this amounts to the passage (a) ⇒ (b), and culminates in the equivalence of all three descriptions (a), (b), (c) of Sturm global attractors It remains to recall the two main concepts mentioned in the above proof of theorem 1.2: meanders M and SZS-pairs (h0, h1) of Hamiltonian paths in C.
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