Structure of the attractor of the Cahn-Hilliard equation on a square

39. Stanislaus Maier-Paape, Konstantin Mischaikow, Thomas Wanner:
    Structure of the attractor of the Cahn-Hilliard equation on a square
    International Journal of Bifurcation and Chaos 17(4), pp. 1221-1263, 2007.

Abstract

We describe the fine structure of the global attractor of the Cahn-Hilliard equation on two-dimensional square domains. This is accomplished by combining recent numerical results on the set of equilibrium solutions due to Maier-Paape and Miller (2002) with algebraic Conley index techniques. Using the information on the set of equilibria as assumption, we build Morse decompositions and connection matrices. The latter imply existence of heteroclinic connections between the equilibria inside the attractor. While path following the parameter range of Cahn-Hilliard, we find more and more complicated dynamical behavior. One of our main results describes the fine structure of the attractor for mean mass zero with four stable cosine structured equilibria and eight other stable equilibria that have a quarter circle nodal line. Besides that, we also study the attractor in symmetry fixed point spaces where we e.g. find nonunique connection matrices and saddle-saddle connections of Morse sets.

Links

The published version of the paper can be found at https://doi.org/10.1142/S0218127407017781.

Bibtex

@article{maier:etal:07a,
   author = {Stanislaus Maier-Paape and Konstantin Mischaikow and
             Thomas Wanner},
   title = {Structure of the attractor of the {C}ahn-{H}illiard
            equation on a square},
   journal = {International Journal of Bifurcation and Chaos},
   year = 2007,
   volume = 17,
   number = 4,
   pages = {1221--1263},
   doi = {10.1142/S0218127407017781}
   }