2018/11/10 by Bernold Fiedler, Fiedler, Bernold, Carlos Rocha +1 · 2 citations
Mathematics · Physics and Astronomy · #05C90 #35B41 #37D15 #57N60 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS #msc:05C90 #msc:35B41 #msc:37D15 #msc:57N60
paper · pdf · doi:10.48550/arxiv.1811.04206
39+(ii) pages, 6 figures
openalex publication_date 2018/11/10 · arxiv created 2020/01/25 · arxiv updated 2020/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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\inE. The global attractor A of \eqrefeq:*, 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\\rightarrowE 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:=h0-1 ∘ h1, conversely, determines the global attractor A uniquely, up to topological conjugacy.