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Abstract

We briefly discuss some classical and some recent results concerning properties of the period in ordinary differential systems of certain types. One is the well known Hopf bifurcation theorem (a particular case) when stability is lost with the variation of a parameter that produces two purely imaginary eigenvalues in the spectrum of the Jacobian matrix calculated at some equilibrium point. Another one is the recent formula for the derivative of the period with respect to perturbations of the Hamiltonian systems in dimension two. These results can be seen as complementary since Hopf theory and its subsequent developments refer mainly to limit cycles, while the Hamiltonian case does not allow the presence of limit cycles, that is, there are no isolated (periodic) solutions. For Hamiltonian systems, the arguments for the differentiability of the period are spread over several papers and we give here a unified and complete treatment. We underline the implicit dependence of the period on the data, which shows the difficulty of such results. The last part of the paper is devoted to applications in shape and topology optimization problems, the case when the state system is given by elliptic variational inequalities with unilateral conditions on the boundary. The Hamiltonian method plays an essential role and the above differentiability properties allow computing the gradient of the cost in this nonlinear and nonsmooth setting.