Is it possible for hamiltonian and gradient systems to have limit cycles or periodic orbits?
I have already proved that gradient systems can't have closed orbits (that means they can't have limit cycles either, right?)
However I am having a harder time proving hamiltonian dynamical systems can/can't have limit cycles or periodic orbits. I know that the Dulac's criterion or existence of Lyapunov functions rule out the existence of closed orbits, and given one dynamical system, one can use the Poincaré-Bendixon theorem to prove a limit cycle exists in a specific region. However that does not help to prove any general case.