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In this work we use standard Hamiltonian-system techniques in order to studythe dynamics of three vortices with alternating charges in a confinedBose-Einstein condensate. In addition to being motivated by recent experiments,this system offers a natural vehicle for the exploration of the transition ofthe vortex dynamics from ordered to progressively chaotic behavior. Inparticular, it possesses two integrals of motion, the {\it energy} (which isexpressed through the Hamiltonian $H$) and the {\it angular momentum} $L$ ofthe system. By using the integral of the angular momentum, we reduce the systemto a two degree-of-freedom one with $L$ as a parameter and reveal the topologyof the phase space through the method of Poincar\'e surfaces of section.  We categorize the various motions that appear in the different regions of thesections and we study the major bifurcations that occur to the families ofperiodic motions of the system. Finally, we correspond the orbits on thesurfaces of section to the real space motion of the vortices in the plane.

Dynamics of Three Non-co-rotating Vortices in Bose-Einstein Condensates