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Bifurcation of point vortex equilibria: four-vortex translating configurations and five-vortex stationary configurations. (English) Zbl 1473.37062

The dynamics of {point vortices} is a classical mathematics playground (see [H. Aref, J. Math. Phys. 48, No. 6, 065401, 23 p. (2007; Zbl 1144.81308)]). One is often interested in equilibria or relative equilibria of point vortices. These are defined by the condition that the vortices (or better to say, the spatial coordinates \({\mathbf x}_{(i)} (t)\) of the vortices \(i =1,\dots,n\)) are either stationary or move with a common constant velocity and hence do not change their relative positions. These solutions have a strong relation with roots of polynomials, and hence with the “Fundamental Theorem of algebra” (see [H. Aref, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 467, No. 2132, 2168–2184 (2011; Zbl 1325.76050)]).
Here the author studies the bifurcations for four-vortex translating configurations and five-vortex stationary configurations. It turns out the number of distinct four-vortex translating configurations can be 0, 3, 4, or 6. Among the nonzero cases, there are 1, 0, and 0, or 2 collinear ones, respectively. The number of distinct five-vortex stationary configurations can be 0, 2, 4, 6. In each case, all smaller or equal even numbers of collinear configurations among them are possible. The paper provides necessary and sufficient conditions on nonzero real circulations for all the possible numbers of collinear and strictly planer equilibrium. Moreover, the merging and the separation of two or three equilibria is described.
All configurations are studied through the minimal polynomial systems provided by K. A. O’Neil [Physica D 219, No. 1, 69–79 (2006; Zbl 1102.76011)]. Many computations are performed by a computer algebra system.

MSC:

37G10 Bifurcations of singular points in dynamical systems
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
37M20 Computational methods for bifurcation problems in dynamical systems
70K42 Equilibria and periodic trajectories for nonlinear problems in mechanics
70K50 Bifurcations and instability for nonlinear problems in mechanics
13P15 Solving polynomial systems; resultants
68W30 Symbolic computation and algebraic computation
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References:

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