STURM 3-BALL GLOBAL ATTRACTORS 3: EXAMPLES OF THOM-SMALE COMPLEXES.
In: Discrete & Continuous Dynamical Systems: Series A, Jg. 38 (2018-07-01), Heft 7, S. 3479-3545
academicJournal
Zugriff:
Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors A which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, ut = uxx + f(x, u, ux) , on the unit interval 0 < x < 1 with Neumann boundary conditions. Equilibria vt = 0 are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the fast unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. In the first two papers we characterized 3-ball Sturm attractors A as 3-cell templates C. The characterization involves bipolar orientations and hemisphere decompositions which are closely related to the geometry of the fast unstable manifolds. An equivalent combinatorial description was given in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries x = 0 and x = 1, respectively, and the overlapping reach of polar serpents in the shooting meander. In the present paper we apply these descriptions to explicitly enumerate all 3-ball Sturm attractors A with at most 13 equilibria. We also give complete lists of all possibilities to obtain solid tetrahedra, cubes, and octahedra as 3- ball Sturm attractors with 15 and 27 equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and dodecahedra, we indicate a reduction to mere planar considerations as discussed in our previous trilogy on planar Sturm attractors. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Continuous Dynamical Systems: Series A is the property of American Institute of Mathematical Sciences and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Titel: |
STURM 3-BALL GLOBAL ATTRACTORS 3: EXAMPLES OF THOM-SMALE COMPLEXES.
|
---|---|
Autor/in / Beteiligte Person: | Fiedler, Bernold ; Rocha, Carlos |
Zeitschrift: | Discrete & Continuous Dynamical Systems: Series A, Jg. 38 (2018-07-01), Heft 7, S. 3479-3545 |
Veröffentlichung: | 2018 |
Medientyp: | academicJournal |
ISSN: | 1078-0947 (print) |
DOI: | 10.3934/dcds.2018149 |
Schlagwort: |
|
Sonstiges: |
|