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 $0 \leq x \leq \pi$ 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 $f = f (u)$. In this class the stationary solutions of the parabolic equation satisfy the second order ODE $v^{\prime\prime} + f (v) = 0$ and we obtain the permutation characterization from a characterization of the set of $2\pi$-periodic orbits of this planar Hamiltonian system. Our results are based on a diligent discussion of this mere pendulum equation