We embark on a detailed analysis of the close relations between combinatorial and geometric aspects of the scalar parabolic PDE

on the unit interval \(0< x<1\) with Neumann boundary conditions. We assume f to be dissipative with N hyperbolic equilibria \(v\in {\mathcal {E}}\). 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

conversely, determines the global attractor \({\mathcal {A}}\) uniquely, up to topological conjugacy.

我们开始对标量抛物线型PDE的组合和几何方面之间的紧密关系进行详细分析

在带有Neumann边界条件的单位间隔\（0 <x <1 \）上。我们假设f对N个双曲均衡\（v \ in {\ mathcal {E}} \）具有耗散性。（*）的全局吸引子\（{\ mathcal {A}} \\），也称为Sturm全局吸引子，由所有均衡v的不稳定流形组成。作为细胞，它们形成了Thom–Smale复数\（{\ mathcal {C}} \\）。基于v的快速不稳定流形，我们引入了规则单元格\（{\ mathcal {C}} \\）的细化\（{\ mathcal {C}} ^ s \），我们称其为有符号Thom–人妖情结。仅给定有符号单元格\（{\ mathcal {C}} ^ s \）及其底层的偏序，我们得出两个总边界阶\（h_ \ iota：\ {1，\ ldots，N \} \在两个诺伊曼边界\（\ iota = x = 0,1 \）处的平衡值v（x）的rightarrow {\ mathcal {E}} \）。在先前的工作中，我们已经确定了结果生成的Sturm排列

相反，唯一确定全局吸引子\（{\ mathcal {A}} \），直到拓扑共轭为止。