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Sturm 3-Ball Global Attractors 2: Design of Thom–Smale Complexes

  • Bernold FiedlerEmail author
  • Carlos Rocha
Article
  • 45 Downloads

Abstract

This is the second 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 is slightly more refined. It 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 first paper we showed the implications
$$\begin{aligned} \text {Sturm attractor}\quad \Longrightarrow \quad \text {Thom}{-}\text {Smale complex} \quad \Longrightarrow \quad \text {meander}\,. \end{aligned}$$
The present part 2 closes the cycle of equivalences by the implication
$$\begin{aligned} \text {meander} \quad \Longrightarrow \quad \text {Sturm attractor}\,. \end{aligned}$$
In particular this cycle allows us to construct a unique Sturm 3-ball attractor for any prescribed Thom–Smale complex which satisfies the geometric properties of the bipolar orientation and the hemisphere decomposition. Many explicit examples and illustrations will be discussed in part 3. The present 3-ball trilogy, however, is just another step towards a still elusive geometric and combinational characterization of all Sturm global attractors in arbitrary dimensions.

Keywords

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

Mathematics Subject Classification

37D15 35B41 05C90 57N60 

Notes

Acknowledgements

We are deeply indebted to George Sell who has encouraged and inspired our work on global attractors for decades. Extended delightful hospitality by the authors is mutually acknowledged. Suggestions concerning the Thom–Smale complex were generously provided by Jean-Michel Bismut. Gustavo Granja has 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

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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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