Boundary Orders and Geometry of the Signed Thom–Smale Complex for Sturm Global Attractors

Abstract

We embark on a detailed analysis of the close relations between combinatorial and geometric aspects of the scalar parabolic PDE ut=uxx+f(x,u,ux) (*)

on the unit interval 0<x<1 with Neumann boundary conditions. We assume f to be dissipative with N hyperbolic equilibria v∈E. The global attractor A of (*), also called Sturm global attractor, consists of the unstable manifolds of all equilibria v. As cells, these form the Thom–Smale complex C. Based on the fast unstable manifolds of v, we introduce a refinement Cs of the regular cell complex C, which we call the signed Thom–Smale complex. Given the signed cell complex Cs and its underlying partial order, only, we derive the two total boundary orders hι:{1,…,N}→E of the equilibrium values v(x) at the two Neumann boundaries ι=x=0,1

. In previous work we have already established how the resulting Sturm permutation σ:=h−10∘h1,

conversely, determines the global attractor A uniquely, up to topological conjugacy.