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We embark on a detailed analysis of the close relations between combinatorial and geometric aspects of the scalar parabolic PDE * $$\begin{aligned} u_t = u_{xx} + f(x,u,u_x) \end{aligned}$$ u t = u xx + f ( x , u , u x ) 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 $${\mathcal {C}}^s$$ C s of the regular cell complex $${\mathcal {C}}$$ C , which we call the signed Thom–Smale complex . Given the signed cell complex $${\mathcal {C}}^s$$ C s and its underlying partial order, only, we derive the two total boundary orders $$h_\iota :\{1,\ldots , N\}\rightarrow {\mathcal {E}}$$ h ι : { 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 : = h 0 - 1 ∘ h 1 , conversely, determines the global attractor $${\mathcal {A}}$$ A uniquely, up to topological conjugacy.

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