Sturm 3-Ball Global Attractors 2: Design of Thom–Smale Complexes

Abstract

This is the second of three papers on the geometric and combinatorial characterization of global Sturm attractors which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$\begin{aligned} u_t = u_{xx} + f(x,u,u_x)\,, \end{aligned}$$ on the unit interval $$0< x<1$$ with Neumann boundary conditions. Equilibria are assumed to be hyperbolic. Geometrically, we study the resulting Thom–Smale dynamic complex with cells defined by the unstable manifolds of the equilibria. The Thom–Smale complex turns out to be a regular cell complex. Our geometric description is slightly more refined. It involves a bipolar orientation of the 1-skeleton, a hemisphere decomposition of the boundary 2-sphere by two polar meridians, and a meridian overlap of certain 2-cell faces in opposite hemispheres. The combinatorial description is 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 first paper we showed the implications $$\begin{aligned} \text {Sturm attractor}\quad \Longrightarrow \quad \text {Thom}{-}\text {Smale complex} \quad \Longrightarrow \quad \text {meander}\,. \end{aligned}$$ The present part 2 closes the cycle of equivalences by the implication $$\begin{aligned} \text {meander} \quad \Longrightarrow \quad \text {Sturm attractor}\,. \end{aligned}$$ In particular this cycle allows us to construct a unique Sturm 3-ball attractor for any prescribed Thom–Smale complex which satisfies the geometric properties of the bipolar orientation and the hemisphere decomposition. Many explicit examples and illustrations will be discussed in part 3. The present 3-ball trilogy, however, is just another step towards a still elusive geometric and combinational characterization of all Sturm global attractors in arbitrary dimensions.