Global determination of a nonlinearity from a periodic motion

We consider a problem of determining a nonlinearity of an autonomous differential equation from a period function, namely, a relation between periods and amplitudes. A global existence and a characterization of nonlinearities realizing a prescribed period function are established under the assumption that the function is Lipschitz continuous. This gives a global answer to a classical inverse problem in a nonlinear autonomous oscillation. We also consider a related problem of determining a nonlinearity from a relation between periods and initial positions with zero velocity, and give a criterion for the nonlinearity to be unique. This result applies to an inverse bifurcation problem.