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)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u_t = u_{xx} + f(x,u,u_x) \\end{aligned}$$\\end{document}on the unit interval 0< x<1 with Neumann boundary conditions. We assume f to be dissipative with N hyperbolic equilibria vin {mathcal {E}}. The global attractor {mathcal {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}}. Based on the fast unstable manifolds of v, we introduce a refinement {mathcal {C}}^s of the regular cell complex {mathcal {C}}, which we call the signed Thom–Smale complex. Given the signed cell complex {mathcal {C}}^s and its underlying partial order, only, we derive the two total boundary orders h_iota :{1,ldots , N}rightarrow {mathcal {E}} of the equilibrium values v(x) at the two Neumann boundaries iota =x=0,1. In previous work we have already established how the resulting Sturm permutation σ:=h0-1∘h1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sigma :=h_{0}^{-1} \\circ h_1, \\end{aligned}$$\\end{document}conversely, determines the global attractor {mathcal {A}} uniquely, up to topological conjugacy.