{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T16:39:08Z","timestamp":1772296748681,"version":"3.50.1"},"reference-count":39,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2007,4]]},"abstract":"<jats:p> We describe the fine structure of the global attractor of the Cahn\u2013Hilliard equation on two-dimensional square domains. This is accomplished by combining recent numerical results on the set of equilibrium solutions due to [Maier-Paape &amp; 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 of the parameter range of Cahn\u2013Hilliard, 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, for example, find nonunique connection matrices and saddle\u2013saddle connections of Morse sets. <\/jats:p>","DOI":"10.1142\/s0218127407017781","type":"journal-article","created":{"date-parts":[[2007,6,12]],"date-time":"2007-06-12T01:09:04Z","timestamp":1181610544000},"page":"1221-1263","source":"Crossref","is-referenced-by-count":27,"title":["STRUCTURE OF THE ATTRACTOR OF THE CAHN\u2013HILLIARD EQUATION ON A SQUARE"],"prefix":"10.1142","volume":"17","author":[{"given":"STANISLAUS","family":"MAIER-PAAPE","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Mathematik, RWTH Aachen, Templergraben 55, 52062 Aachen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"KONSTANTIN","family":"MISCHAIKOW","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghusen Rd, Piscataway, NJ 08854-8019, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"THOMAS","family":"WANNER","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","volume-title":"Attractors of Evolution Equations","author":"Babin A. 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