Chaotic scattering of two identical point vortex pairs

Abstract

The scattering dynamics of two vortex pairs with equal circulations ±κ have hitherto been thought to be regular, in contrast to the case of nonidentical pairs where a hierarchy of resonant states gives rise to chaotic behavior. However, no rigorous explanation could be found for such regularity occurring in a nonintegrable Hamiltonian system. This was recently posed as an open problem [H. Aref et al., Fluid Dyn. Res. 3, 63 (1988)], and in this paper the problem is resolved by presenting a counter-example mechanism which yields chaotic scattering dynamics when the initial pair spacings differ markedly. Alternative modes of scattering are characterized by symbolic sequences. These predictions are confirmed by numerical experiments which also show that the chaos persists even when the initial spacings are nearly equal, although the physical origin of the irregular scattering is less clear. Regular behavior is seen only in the integrable case of equal initial pair spacings.