Mathematical and numerical analysis to the assembly of vortices in fluid

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(1) We consider five point vortices on the vertices and center of a rectangular. We already known that there exists a periodic motion of vortices for some parameters on the initial configuration and strength of vortices. We study the periodic motion for another value of parameters. As a result, we find that there exists a family of periodic motions. By numerical simulations, these periodic motions have a simple structure.(2) We consider five point vortices on the vertices and center of a diamond. For some values of strength of vortices, the five vortices is in a relative equilibrium, that is, it is in a equilibrium with respect to a rotation coordinate with uniform angular velocity. There are two parameters in this problem. We study the stability of the relative equilibrium. By numerical simulations, we find that the equilibria are stable in a narrow parameter range. To show the stability, we construct the Lyapunouv function and apply the computer-assisted proof using interval computations. As a result, under the same signature of strengths, we find that many elliptic equilibria are stable. Until now we do not succeed in proving stability for all elliptic equilibria.(3) Under the same situation stated in (2), we numerically find that some unstable equilibria exhibit the relaxation oscillation. We do not know whether or not all unstable equilibria exhibit the oscillation, however, our numerical simulations suggest that an unstable equilibrium which does not exhibit the oscillation exists. We already know that there are square-shaped five point vortices which exhibit the oscillation, however, the oscillation we find belongs to a different kind category. The mathematical reason why such a oscillation occurs is one of our future works.