I am currently interested in the study of periodic solutions for the N-vortex problem. The N-vortex system has 160 years of history. It was first formulated by Helmholtz in 1868 and the Hamiltonian structure of the dynamic system is investigated by Kirchhoff in 1876. As a Hamiltonian system, it is in general not Liouville integrable when N>3, while regular and chaotic behaviours co-exist. It could be seen as a finite dimensional approximation for turbulence of flows from hydrodynamics ( for example the Euler equation ) and quantum mechanics ( for example the Gross-Pitaevskii equation ). Due to the non-integrability, one should not expect a completely explicit description of its behaviours in general. In this case the search of periodic solutions, as suggested by Henri Poincaré, might be:
"the only opening through which we can try to penetrate in a place which, to now, was supposed to be inaccessible."
--- <Les nouvelles méthodes de la mécanique céleste>