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April  2005, 12(3): 531-554. doi: 10.3934/dcds.2005.12.531

Describing a class of global attractors via symbol sequences

Jörg Härterich 1, and  Matthias Wolfrum 2,

1. 

Institut für Mathematik I, Freie Universität Berlin, Arnimallee 2-6, 14195 Berlin, Germany
2. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

Received  May 2003 Revised  October 2004 Published  December 2004

We study a singularly perturbed scalar reaction-diffusion equation on a bounded interval with a spatially inhomogeneous bistable nonlinearity. For certain nonlinearities, which are piecewise constant in space on $k$ subintervals, it is possible to characterize all stationary solutions for small $\varepsilon$ by means of sequences of $k$ symbols, indicating the behavior of the solution in each subinterval. Determining also Morse indices and zero numbers of the equilibria in terms of the symbol sequences, we are able to give a criterion for heteroclinic connections and a description of the associated global attractor for all $k$.
Keywords: heteroclinic orbit, Singular perturbation, transition layer, global attractor, spike solution..
Mathematics Subject Classification: 35B40, 35B41, 35K57, 34E15, 37C29, 37L3.
Citation: Jörg Härterich, Matthias Wolfrum. Describing a class of global attractors via symbol sequences. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 531-554. doi: 10.3934/dcds.2005.12.531
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