Consider the Hamiltonian system \((*)\quad J\dot z=H'(z)\), \(z\in \mathbb{R}^{2N}\), where the Hamiltonian \(H\) is convex and superquadratic at the origin and infinity. According to the Clarke-Ekeland duality principle, \(T\)-periodic solutions of \((*)\) correspond to critical points of a certain dual functional. Moreover, by a result of Ekeland and Hofer, there exists a critical point of mountain pass type and the corresponding solution has minimal period \(T\).
This paper is concerned with a family of systems \((*)_n\) \(J\dot z=H_n'(z)\), where the \(H_n\)’s are as above and \(H_n\to H\) in a suitable sense. It is shown that \(T\)-periodic solutions \(z_n\) of \((*)_n\) corresponding to mountain pass points have a subsequence converging to a solution \(z\) of \((*)\). Moreover, the minimal period of \(z\) is \(T\) or \(T/2\).
The proof makes use of the Morse index estimates for the dual functional. A similar problem is also considered for a class of Hamiltonian systems with \(H\) convex, quadratic at the origin and superquadratic at infinity.