While studying a physical system I've encountered an two dimensional autonomous ODE defined on a torus. The vector field describing it is a messy expression involving rational functions of trigonometric functions (it's smooth).
Looking at a phase portrait it seems like the dynamics could be Hamiltonian (image attached) which would be a significant result for the physical problem at hand.
Is there a test that allows to verify if a system in 2D is Hamiltonian?
Something like computing divergence or curl of the field is feasible, but integrating trajectories analytically is not feasible.
If not, negative answer with some examples would be greatly appreciated.
