Journal of Nonlinear Science

, Volume 27, Issue 3, pp 775–815 | Cite as

Numbers of Relative Equilibria in the Planar Four-Vortex Problem: Some Special Cases



Three planar four-vortex problems are considered in this paper. In the \((3+1)\)-vortex problem, we study the relative equilibria of the four point vortices when one vortex has zero vorticity and the other three with nonzero vorticities form an equilateral triangle. In the \((1+3)\)-vortex problem, we study the limiting cases of the relative equilibria when one of the four point vortices has fixed nonzero vorticity and other vorticities approach zero. The third problem is the case of vanishing total vorticity. All problems involve two real vorticity parameters. We consider all meaningful pairs of parameters and find there can only be 4, 8, 9 or 10 relative equilibria in the \((3+1)\)-vortex problem, and 8, 10, 12 or 14 relative equilibria in the \((1+3)\)-vortex problem. For the case of zero total vorticity, there are 0, 1 or 2 collinear relative equilibria and 2, 3 or 4 strictly planar relative equilibria. We completely classify parameters according to the different numbers of relative equilibria. For all cases, we reduce them to the problems of counting common zeros in an open region of \({{{\mathbb {R}}}}^{2}\) for polynomial systems with two equations, two variables, and two parameters. We propose a method to count zeros for such type of systems for all parameters in an open region of \({\mathbb {R}}^{2}\) through symbolic computations. Therefore, all of our results are proved rigorously.


Four point vortices Relative Equilibria Parametric polynomial systems Real root counting 



The author would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. This research was partly supported by the Ministry of Science and Technology of Taiwan under the Grant MOST 105-2115-M-005-006.


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Chung Hsing UniversityTaichung CityTaiwan

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