# Existence of limit cycles and periodic orbits in hamiltonian systems.

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.

• From the context I guess that you mean $1$ degree-of-freedom Hamiltonian systems. For such systems there can be closed orbits (take a pendulum equation, general or for small amplitude) but no limit cycles (otherwise, the Hamiltonian would be constant, at lest on one side of the cycle). – user539887 Mar 25 at 7:39
• Could you elaborate on why they can't have limit cycles? I don't understand your reasoning. Thank you for your comment! – codingnight Mar 25 at 9:15
• If a two-dimensional system of ODEs has a limit cycle, then for any point in some (at least one-sided) nbhd of the limit cycle its forward trajectory converges, as $t\to\infty$, to the limit cycle. It follows then that any first integral is constant on that one-sided nbhd. And the Hamiltonian function is a first integral. – user539887 Mar 25 at 9:40