Numerical and Mathemtical Analysis of the motion of vortices

(1)考虑二维欧拉流体中的五点涡。设α和κ分别为旋涡初始构型和旋涡强度的参数。当α=1且κ<-0.5时，我们的数值模拟显示涡旋的旋转运动出现松弛振荡。从数学上证明了引起这种运动的异斜轨道的存在。我们还证明了在α=1和κ>-0.5的扰动下旋转运动是稳定的。当α≠1时，我们发现涡旋具有周期性或拟周期性。在一定情况下，我们证明了周期运动的存在。通过数值模拟，我们指出了周期运动发生的α和κ值。我们还分析了周期运动的形状。(2)采用轮廓动力学方法对五种有限涡的运动进行了数值模拟。对于一定的面积，我们的模拟表明，有限的涡旋开始变形和快速旋转。正如许多研究者已经报道的那样，在较大的区域内，观察到涡旋的合并。我们还对流体上的有限涡和点涡进行了模拟。(3)考虑由周期性五点涡诱导的被动平流粒子的运动。我们的分析是基于粒子在庞加莱截面上运动的数值模拟。我们发现，根据粒子的初始位置，有以下几种情况。粒子停留在漩涡附近。(2)粒子移动远区。(3)粒子出现混沌行为。(4)截面上的颗粒集中在某一区域。

(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.