A permutation characterization of Sturm attractors of Hamiltonian type

We consider Neumann boundary value problems of the form u_t = u_xx + f on the interval leq x leq pi$ for dissipative nonlinearities f = f (u). A permutation characterization for the global attractors of the semiflows generated by these equations is well known, even in the general case f = f (x, u, u_x ). We present a permutation characterization for the global attractors in the restrictive class of nonlinearities f = f (u) this class the stationary solutions of the parabolic equation satisfy the second order ODE v^primeprime + f (v) = 0 and we obtain the permutation characterization from a characterization of the set of 2pi-periodic orbits of this planar Hamiltonian system. Our results are based on a diligent discussion of this mere pendulum equation