Transition from quasiperiodicity to chaos for three coaxial vortex rings

The dynamics of three coaxial vortex rings of strengths Gamma(1), Gamma(2) and Gamma(3) in an ideal fluid is investigated. It is proved that if Gamma(j), Gamma(j) + Gamma(k) and Gamma(1) + Gamma(2) + Gamma(3) are not zero for all j, k = 1, 2, 3, then KAM and Poincare-Birkhoff theory can be used to prove that if the distances among the rings are sufficiently small compared to the mean radius of the rings, there are many initial configurations of the rings that produce quasiperiodic or periodic motions. Moreover, it is shown that the motion become chaotic as the inter-ring distances are increased relative to the mean radius.