Numerical and Mathemtical Analysis of the motion of vortices

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(1)We consider five point vortices in the two-dimensional Euler fluid. Let α and κ be parameters which indicate the initial configulation of the vortices and the strength of a vortex, respectively. When α=1 and κ<-0.5, our numerical simulation show the rotating motion of vortices with relaxation oscillations appears. Mathematically we prove the existence of the heteroclinic orbits, which induces such a motion. We also prove that the rotating motion is stable against some perturbation for α=1 and κ>-0.5. When α≠1, we find that the vortices behave periodic or quasi-periodic. Under certain situation, we prove that periodic motion occurs. By numerical simulations, we indicate the values of α and κ under which periodic motion occurs. We also analyze the shape of the periodic motion.(2)We make numerical simulations for the motion of five finite vortices by the contour dynamics method. For some value of the area, our simulations display that the finite vortices begin to deform and rotate rapidly. For large value of the area, as many researchers are already reported, the coalescence of vortices is observed. We also make simulations for finite and point vortices are on the fluid.(3)We consider the motion of passively advected particles in the flow induced by five point vortices which behave periodic. Our analysis is based on the numerical simulations of the motion of particles on the Poincare section. We find that, according to the initial position of particle, the following cases occurs. (1)The particle stays near the vortex. (2)The particle moves the far area. (3)Chaotic behavior of the paricle occurs. (4)The particle on the section is concentrated at some area.