Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field

Abstract

In this study, we geometrically analyze the relation between a point vortex system and deviation curvatures on the Jacobi field. First, eigenvalues of deviation curvatures are calculated from relative distances of point vortices in a three point vortex system. Afterward, based on the assumption of self-similarity, time evolutions of eigenvalues of deviation curvatures are shown. The self-similar motions of three point vortices are classified into two types, expansion and collapse, when the relative distances vary monotonously. Then, we find that the eigenvalues of self-similarity are proportional to the inverse fourth power of relative distances. The eigenvalues of the deviation curvatures monotonically convergent to zero for expansion, whereas they monotonically diverge for collapse, which indicates that the strengths of interactions between point vortices related to the time evolution of spatial geometric structure in terms of the deviation curvatures. In particular, for collapse, the collision point becomes a geometric singularity because the eigenvalues of the deviation curvature diverge. These results show that the self-similar motions of point vortices are classified by eigenvalues of the deviation curvature. Further, nonself-similar expansion is numerically analyzed. In this case, the eigenvalues of the deviation curvature are nonmonotonous but converge to zero, suggesting that the motion of the nonself-similar three point vortex system is also classified by eigenvalues of the deviation curvature.

Application of Jacobi stability analysis to a first-order dynamical system: relation between nonlinearizability of one-dimensional differential equation and Jacobi stable region

“…Then, a new second-order differential equation was created by considering the solutions of the first-order, onedimensional system as a time potential, and the catastrophic shift of the second-order differential equations was analyzed in terms of the fifth invariant. In [26], second-order differential equations of three-point vortices were obtained by differentiating both sides of the first-order differential equations. Additionally, the behavior of the point vortex was classified using the second invariant.…”

“…When [26] was studied, we noticed that the deviation curvatures of first-order one-dimensional differential equations are calculated by two methods as follows: Method 1 is only differentiating both sides of the equation. Method 2 is differentiating both sides of the equation and then substituting the original equation into the second-order system.…”

“…Method 2 is differentiating both sides of the equation and then substituting the original equation into the second-order system. In the study [26], the relation between the solution of the original equation and the deviation curvature was indicated using only Method 1 because the three-dimensional differential equations are complicated to investigate the difference between these methods. In cases of three-or higher-dimensional differential equations, there are multiple eigenvalues of the deviation curvature, and it is difficult to find them analytically and compare Methods 1 and 2.…”