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We consider the motion of N vortex points on sphere, called the N-vortex problem, which is a Hamiltonian dynamical system. The three-vortex problem is integrable and its motion has already been resolved. On the other hand, when the moment of vorticity vector, which consists of weighed sums of the vortex positions, is zero at the initial moment, the four-vortex problem is integrable, but it has not been investigated yet. The present paper gives a description of the integrable four-vortex problem with the reduction method to a three-vortex problem used by Aref and Stremler. Moreover, we examine whether the vortex points collide self-similarly in finite time. The four-vortex collapse is proved to be impossible. We consider if it is possible for not all but part of the vortex points to collapse self-similarly. Moreover, we discuss the topological structure of periodic orbits obtained in the present problem. [ABSTRACT FROM AUTHOR]

We consider the particle mixing in the plane by two vortex points appearing one after the other, called the blinking vortex system. Mathematical and numerical studies of the system reveal that the chaotic particle mixing, i.e., the chaotic advection, is observed due to the homoclinic chaos, but the mixing region is restricted locally in the neighborhood of the vortex points. The present article shows that it is possible to realize a global and efficient chaotic advection in the blinking vortex system with the help of the Thurston–Nielsen theory, which classifies periodic orbits for homeomorphisms in the plane into three types: periodic, reducible, and pseudo-Anosov (pA). It is mathematically shown that periodic orbits of pA type generate a complicated dynamics, which is called topological chaos. We show that the combination of the local chaotic mixing due to the topological chaos and the dipole-like return orbits realize an efficient and global particle mixing in the blinking vortex system. [ABSTRACT FROM AUTHOR]