# American Institute of Mathematical Sciences

December  2012, 7(4): 617-659. doi: 10.3934/nhm.2012.7.617

## Sturm global attractors for $S^1$-equivariant parabolic equations

 1 Freie Universität Berlin, Institut für Mathematik I, Arnimallee 2-6, D-14195 Berlin, Germany 2 Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Departamento de Matemática, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal, Portugal

Received  January 2012 Published  December 2012

We consider a semilinear parabolic equation of the form $u_t = u_{xx} + f(u,u_x)$ defined on the circle $x ∈ S^1=\mathbb{R}/2\pi\mathbb{Z}$. For a dissipative nonlinearity $f$ this equation generates a dissipative semiflow in the appropriate function space, and the corresponding global attractor $A_f$ is called a Sturm attractor. If $f=f(u,p)$ is even in $p$, then the semiflow possesses an embedded flow satisfying Neumann boundary conditions on the half-interval $(0,\pi)$. This is due to $O(2)$ equivariance of the semiflow and, more specifically, due to reflection at the axis through $x=0,\pi\in S^1$. For general $f=f(u,p)$, where only $SO(2)$ equivariance prevails, we will nevertheless use the Sturm permutation $\sigma$ introduced for the characterization of Neumann flows to obtain a purely combinatorial characterization of the Sturm attractors $A_f$ on the circle. With this Sturm permutation $\sigma$ we then enumerate and describe the heteroclinic connections of all Morse-Smale attractors $A_f$ with $m$ stationary solutions and $q$ periodic orbits, up to $n:=m+2q \le 9$.
Citation: Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
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