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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


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Volume 2009, Issue 635

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Connectivity and design of planar global attractors of Sturm type, I: Bipolar orientations and Hamiltonian paths

Bernold Fiedler
  • Institut für Mathematik, Freie Universität Berlin, Arnimallee 2-6, 14195 Berlin, Germany. e-mail:
  • Other articles by this author:
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/ Carlos Rocha
Published Online: 2009-07-30 | DOI: https://doi.org/10.1515/CRELLE.2009.076

Abstract

Based on a Morse-Smale structure we study planar global attractors of the scalar reaction-advection-diffusion equation ut = uxx + ƒ(x, u, ux ) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of ƒ, and hyperbolicity of equilibria. We call Sturm attractor because our results strongly rely on nonlinear nodal properties of Sturm type.

The planar Sturm attractor consists of equilibria of Morse index 0, 1, or 2, and their heteroclinic connecting orbits. The unique heteroclinic orbits between adjacent Morse levels define a plane graph which we call the connection graph. Its 1-skeleton consists of the unstable manifolds (separatrices) of the index-1 Morse saddles.

We present two results which completely characterize the connection graphs and their 1-skeletons , in purely graph theoretical terms. Connection graphs are characterized by the existence of pairs of Hamiltonian paths with certain chiral restrictions on face passages. Their 1-skeletons are characterized by the existence of cycle-free orientations with only one maximum and only one minimum. Such orientations are called bipolar in [de Fraysseix, de Mendez, Rosenstiehl, Discr. Appl. Math. 56: 157–179, 1995].

In the present paper we show the equivalence of the two characterizations. Moreover we show that connection graphs of Sturm attractors indeed satisfy the required properties. In [Fiedler and Rocha, J. Diff. Equ. 244: 1255–1286, 2008] we show, conversely, how to design a planar Sturm attractor with prescribed plane connection graph or 1-skeleton of the required properties. In [Fiedler and Rocha, Connectivity and design of planar global attractors of Sturm type, III: Small and Platonic examples, 2008] we describe all planar Sturm attractors with up to 11 equilibria. We also design planar Sturm attractors with prescribed Platonic 1-skeletons.

About the article

Received: 2007-05-03

Revised: 2007-11-30

Published Online: 2009-07-30

Published in Print: 2009-10-01


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2009, Issue 635, Pages 71–96, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/CRELLE.2009.076.

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[1]
Bernold Fiedler and Carlos Rocha
Journal of Dynamics and Differential Equations, 2010, Volume 22, Number 2, Page 121
[2]
Anna Karnauhova and Stefan Liebscher
Discrete and Continuous Dynamical Systems, 2017, Volume 37, Number 9, Page 4835
[3]
Carlos Rocha and Bernold Fiedler
Discrete and Continuous Dynamical Systems, 2014, Volume 34, Number 12, Page 5099
[4]
Matthias Wolfrum, Carlos Rocha, and Bernold Fiedler
Networks and Heterogeneous Media, 2012, Volume 7, Number 4, Page 617

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