We consider the set of 2π-periodic solutions of the ordinary differential equation u′′ + g(u) = 0 for a nonlinearity \documentclass[12pt]{minimal}
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\begin{document}$$g \in C^1(\mathbb{R})$$\end{document}, satisfying a dissipative condition of the form \documentclass[12pt]{minimal}
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\begin{document}$$g(u) /u < 0$$\end{document} for \documentclass[12pt]{minimal}
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\begin{document}$$|u| > M$$\end{document} , and under the generic assumption that the potential G, given by \documentclass[12pt]{minimal}
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\begin{document}$$G(u)=\int_0^u g(s) ds$$\end{document}, is a Morse function. Under these assumptions, we characterize the period maps realizable by planar Hamiltonian systems of the form \documentclass[12pt]{minimal}
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\begin{document}$$u^{\prime\prime}+g(u)=0$$\end{document} . Considering the Morse type of G, the set of periodic orbits in the phase space \documentclass[12pt]{minimal}
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\begin{document}$$(u,u^\prime)$$\end{document} 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 2π-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 \documentclass[12pt]{minimal}
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\begin{document}$$u_t = u_{xx}+f(u,u_x)$$\end{document} defined on the circle, \documentclass[12pt]{minimal}
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\begin{document}$$x \in S^1$$\end{document} .
