Realization of period maps of planar Hamiltonian systems

We consider the set of 2 pi-periodic solutions of the ordinary differential equation u '' + g(u)= 0 for a nonlinearity g epsilon C-1(R), satisfying a dissipative condition of the form g(u)/u < 0 for |u| > M, and under the generic assumption that the potential G, given by G(u)=integral(u)(0)g(s)ds, is a Morse function. Under these assumptions, we characterize the period maps realizable by planar Hamiltonian systems of the form u '' + g(u) = 0. Considering the Morse type of G, the set of periodic orbits in the phase space (u, u '') is decomposed into disks and annular regions. Then, the realizable period maps are described in terms of sets of sequences of positive integers corresponding to the lap numbers of the pi-periodic solutions. This leads to a characterization of the classes of Morse-Smale attractors that are realizable by dissipative semilinear parabolic equations of the form u(t) = u(xx), + f(u, u(x)) defined on the circle, x epsilon S-1.