Sturm 3-ball global attractors 1: Thom-Smale complexes and meanders

Abstract: This is the first 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, $$ u_t = u_{xx} + f(x,u,u_x)\,, $$ 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 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 present paper we show the implications $$ \text{Sturm attractor}\quad \Longrightarrow \quad \text{Thom-Smale complex} \quad \Longrightarrow \quad \text{meander}\,.$$ The sequel, part 2, closes the cycle of equivalences by the implication $$ \text{meander} \quad \Longrightarrow \quad \text{Sturm attractor}\,.$$ Many explicit examples will be discussed in part 3. The present 3-ball trilogy extends our previous trilogy on planar Sturm global attractors towards the still elusive goal of geometric and combinational characterizations of all Sturm global attractors of arbitrary dimension.