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Sturm 3-ball global attractors 1: Thom–Smale complexes and meanders

  • Bernold FiedlerEmail author
  • Carlos Rocha
Article

Abstract

This is the first of three papers on the geometric and combinatorial characterization of global Sturm attractors which consist of a single closed 3-ball. The underlying scalar PDE is parabolic,
$$\begin{aligned} u_t = u_{xx} + f(x,u,u_x), \end{aligned}$$
on the unit interval \(0< x<1\) with Neumann boundary conditions. Equilibria are assumed to be hyperbolic. Geometrically, we study the resulting Thom–Smale dynamic complex with cells defined by the unstable manifolds of the equilibria. The Thom–Smale complex turns out to be a regular cell complex. Our geometric description involves a bipolar orientation of the 1-skeleton, a hemisphere decomposition of the boundary 2-sphere by two polar meridians, and a meridian overlap of certain 2-cell faces in opposite hemispheres. The combinatorial description is 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 show the implications
$$\begin{aligned} \text {Sturm attractor}\quad \Longrightarrow \quad \text {Thom--Smale complex} \quad \Longrightarrow \quad \text {meander}. \end{aligned}$$
The sequel, part 2, closes the cycle of equivalences by the implication
$$\begin{aligned} \text {meander} \quad \Longrightarrow \quad \text {Sturm attractor}. \end{aligned}$$
Each implication, or mapping, involves certain constructions which are tuned such that the final 3-ball Sturm global attractor defined by the meander combinatorics coincides with the originally given Sturm 3-ball. Many explicit examples and illustrations will be discussed in part 3. The present 3-ball trilogy extends our previous trilogy on planar Sturm global attractors towards the still elusive goal of geometric and combinational characterizations of all Sturm global attractors of arbitrary dimension.

Keywords

Parabolic partial differential equation Nodal property Regular cell complex Bipolar graph Hamiltonian path 

Notes

Acknowledgements

With great pleasure we express our profound gratitude to Waldyr M. Oliva, whose deep geometric insights and friendly challenges are a visible inspiration for us since so many years. Extended delightful hospitality by the authors is mutually acknowledged. Suggestions concerning the Thom–Smale complex were generously provided by Jean–Michel Bismut. Gustavo Granja generously shared his deeply topological view point, precise references included. Anna Karnauhova has contributed all illustrations with great patience, ambition, and her inimitable artistic touch. Typesetting was expertly accomplished by Ulrike Geiger. This work was partially supported by DFG/Germany through SFB 910 project A4 and by FCT/Portugal through project UID/MAT/04459/2013.

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Copyright information

© Instituto de Matemática e Estatística da Universidade de São Paulo 2017

Authors and Affiliations

  1. 1.Institut für MathematikFreie Universität BerlinBerlinGermany
  2. 2.Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal

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