Meanders, zero numbers and the cell structure of Sturm global attractors

Abstract

We study global attractors $\mathcal{A}={\mathcal{A}}_f$ of semiflows generated by semilinear partial parabolic differential equations of the form $u_t=u_{xx}+f(x,u,u_x),0<x<1$, satisfying Neumann boundary conditions. The equilibria $v\in \mathcal{E}\subset \mathcal{A}$ of the semiflow are the stationary solutions of the PDE, hence they are solutions of the corresponding second order ODE boundary value problem. Assuming hyperbolicity of all equilibria, the dynamic decomposition of $\mathcal{A}$ into unstable manifolds of equilibria provides a geometric and topological characterization of Sturm global attractors $\mathcal{A}$ as finite regular signed CW-complexes, the Sturm complexes, with cells given by the unstable manifolds of equilibria. Concurrently, the permutation $\sigma ={\sigma }_f$ derived from the ODE boundary value problem by ordering the equilibria according to their values at the boundaries $x=0,1$, respectively, completely determines the Sturm global attractor $\mathcal{A}$. Equivalently, we use a planar curve, the meander $\mathcal{M}={\mathcal{M}}_f$, associated to the the ODE boundary value problem by shooting. The main objective of this paper is to derive a minimax property which identifies the equilibria on the cell boundary of $\mathcal{O}$ which are closest or most distant from $\mathcal{O}$ at the boundaries $x=0,1$, directly from the permutation $\sigma$, the Sturm permutation, or equivalently from the meander $\mathcal{M}$, the Sturm meander, based on the Sturm nodal properties of the solutions of the ODE boundary value problem. We emphasize the local aspect of this result by applying it to an example for which the identification of the equilibria in the cell boundary of $\mathcal{O}$ is obtained from the knowledge of only a section of the Sturm meander $\mathcal{M}$.

27 pages, 9 figures. Dedicated to the memory of Pavol Brunovsky