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

Abstract

AbstractWe embark on a detailed analysis of the close relations between combinatorial and geometric aspects of the scalar parabolic PDE $$\begin{aligned} u_t = u_{xx} + f(x,u,u_x) \end{aligned}$$ut=uxx+f(x,u,ux)on the unit interval $$0< x<1$$0<x<1 with Neumann boundary conditions. We assume f to be dissipative with N hyperbolic equilibria $$v\in {\mathcal {E}}$$v∈E. The global attractor $${\mathcal {A}}$$A of (*), also called Sturm global attractor, consists of the unstable manifolds of all equilibria v. As cells, these form the Thom–Smale complex$${\mathcal {C}}$$C. Based on the fast unstable manifolds of v, we introduce a refinement $${\mathcal {C}}^s$$Cs of the regular cell complex $${\mathcal {C}}$$C, which we call the signed Thom–Smale complex. Given the signed cell complex $${\mathcal {C}}^s$$Cs and its underlying partial order, only, we derive the two total boundary orders $$h_\iota :\{1,\ldots , N\}\rightarrow {\mathcal {E}}$$hι:{1,…,N}→E of the equilibrium values v(x) at the two Neumann boundaries $$\iota =x=0,1$$ι=x=0,1. In previous work we have already established how the resulting Sturm permutation $$\begin{aligned} \sigma :=h_{0}^{-1} \circ h_1, \end{aligned}$$σ:=h0-1∘h1,conversely, determines the global attractor $${\mathcal {A}}$$A uniquely, up to topological conjugacy.