A permutation characterization of Sturm global attractors of Hamiltonian type

We consider Neumann boundary value problems of the form ut=uxx+f on the interval 0⩽x⩽π 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 much more general case f=f(x,u,ux). 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″+f(v)=0 and we obtain the permutation characterization from a characterization of the set of 2π-periodic orbits of this planar Hamiltonian system. Our results are based on a diligent discussion of this mere pendulum equation.