Orbit Equivalence of Global Attractors of Semilinear Parabolic Differential Equations

We consider global attractors Af of dissipative parabolic equations ut = uxx + f(x, u, ux) on the unit interval 0 ≤ x ≤ 1 with Neumann boundary conditions. A permutation πf is defined by the two orderings of the set of (hyperbolic) equilibrium solutions ut ≡ 0 according to their respective values at the two boundary points x = 0 and x = 1. We prove that two global attractors, Af and Ag, are globally C0 orbit equivalent, if their equilibrium permutations πf and πg coincide. In other words, some discrete information on the ordinary differential equation boundary value problem ut ≡ 0 characterizes the attractor of the above partial differential equation, globally, up to orbit preserving homeomorphisms.