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We assume f<\/jats:italic> to be dissipative with N<\/jats:italic> hyperbolic equilibria $$v\\in {\\mathcal {E}}$$<\/jats:tex-math>\n \n v<\/mml:mi>\n \u2208<\/mml:mo>\n E<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The global attractor $${\\mathcal {A}}$$<\/jats:tex-math>\n A<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of (*), also called Sturm global attractor<\/jats:italic>, consists of the unstable manifolds of all equilibria v<\/jats:italic>. As cells, these form the Thom\u2013Smale complex<\/jats:italic>$${\\mathcal {C}}$$<\/jats:tex-math>\n C<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Based on the fast unstable manifolds of v<\/jats:italic>, we introduce a refinement $${\\mathcal {C}}^s$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of the regular cell complex $${\\mathcal {C}}$$<\/jats:tex-math>\n C<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, which we call the signed Thom\u2013Smale complex<\/jats:italic>. Given the signed cell complex $${\\mathcal {C}}^s$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and its underlying partial order, only, we derive the two total boundary orders $$h_\\iota :\\{1,\\ldots , N\\}\\rightarrow {\\mathcal {E}}$$<\/jats:tex-math>\n \n \n h<\/mml:mi>\n \u03b9<\/mml:mi>\n <\/mml:msub>\n :<\/mml:mo>\n \n {<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n N<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n \u2192<\/mml:mo>\n E<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of the equilibrium values v<\/jats:italic>(x<\/jats:italic>) at the two Neumann boundaries $$\\iota =x=0,1$$<\/jats:tex-math>\n \n \u03b9<\/mml:mi>\n =<\/mml:mo>\n x<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In previous work we have already established how the resulting Sturm permutation $$\\begin{aligned} \\sigma :=h_{0}^{-1} \\circ h_1, \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \u03c3<\/mml:mi>\n :<\/mml:mo>\n =<\/mml:mo>\n \n h<\/mml:mi>\n \n 0<\/mml:mn>\n <\/mml:mrow>\n \n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n \u2218<\/mml:mo>\n \n h<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula>conversely, determines the global attractor $${\\mathcal {A}}$$<\/jats:tex-math>\n A<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> uniquely, up to topological conjugacy.<\/jats:p>","DOI":"10.1007\/s10884-020-09836-5","type":"journal-article","created":{"date-parts":[[2020,4,16]],"date-time":"2020-04-16T09:03:39Z","timestamp":1587027819000},"page":"2787-2818","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Boundary Orders and Geometry of the Signed Thom\u2013Smale Complex for Sturm Global Attractors"],"prefix":"10.1007","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1141-8475","authenticated-orcid":false,"given":"Bernold","family":"Fiedler","sequence":"first","affiliation":[]},{"given":"Carlos","family":"Rocha","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,4,16]]},"reference":[{"key":"9836_CR1","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1016\/0022-0396(86)90093-8","volume":"62","author":"S Angenent","year":"1986","unstructured":"Angenent, S.: The Morse\u2013Smale property for a semi-linear parabolic equation. 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