Dynamics of three vortices on a plane and a sphere -- III. Noncompact case. Problems of collaps and scattering

Abstract

In this article we considered the integrable problems of three vortices on a plane and sphere for noncompact case. We investigated explicitly the problems of a collapse and a scattering of vortices and obtained the conditions of its realization. We completed the bifurcation analysis and investigated for collinear and Thomson's configurations the dependence of stability in linear approximation and frequency of rotation in relative coordinates from value of a full moment. We indicated the geometric interpretation for characteristic situations. We constructed a phase portrait and geometric projection for the integrable configuration of four vortices on a plane.