The motion of fluids can sometimes be effectively modeled by a finite-dimensional dynamical system. A paradigmatic example is the point vortex model, introduced by Helmholtz in a seminal paper, which describes a regime where the vorticity of an ideal, incompressible fluid is confined to a set of infinitely thin, straight, parallel vortex filaments. In this talk, I will present recent results concerning an incompressible, inviscid fluid with axial symmetry and no swirl, establishing a connection between the motion of a system of N concentrated vortex rings with large radii and a dynamical system closely related to the point vortex model. Based on joint work with Guido Cavallaro and Carlo Marchioro.