# American Institute of Mathematical Sciences

September  2019, 11(3): 427-438. doi: 10.3934/jgm.2019021

## Relative periodic solutions of the $n$-vortex problem on the sphere

 Depto. Matemáticas y Mecánica IIMAS, Universidad Nacional Autónoma de México, Apdo. Postal 20-726, 01000 Ciudad de México, México

Received  September 2018 Revised  April 2019 Published  August 2019

Fund Project: This project is supported by PAPIIT-UNAM grant IN115019.

This paper gives an analysis of the movement of $n\$vortices on the sphere. When the vortices have equal circulation, there is a polygonal solution that rotates uniformly around its center. The main result concerns the global existence of relative periodic solutions that emerge from this polygonal relative equilibrium. In addition, it is proved that the families of relative periodic solutions contain dense sets of choreographies.

Citation: Carlos García-Azpeitia. Relative periodic solutions of the $n$-vortex problem on the sphere. Journal of Geometric Mechanics, 2019, 11 (3) : 427-438. doi: 10.3934/jgm.2019021
##### References:
 [1] Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, 1. American Institute of Mathematical Sciences (AIMS), 2006. [2] T. Bartsch, Topological Methods for Variational Problems with Symmetries, Lecture Notes in Mathematics 1560. Springer-Verlag, 1993. doi: 10.1007/BFb0073859. [3] T. Bartsch and Q. Dai, Periodic solutions of the N-vortex Hamiltonian system in planar domains,, J. Differential Equations, 260 (2016), 2275-2295.  doi: 10.1016/j.jde.2015.10.002. [4] S. Boatto and H. Cabral, Nonlinear stability of a latitudinal ring of point-vortices on a nonrotating sphere,, SIAM J. Appl. Math., 64 (2003), 216-230.  doi: 10.1137/S0036139902399965. [5] V. A. Bogolmonov, Dynamics of vorticity at a sphere, Fluid. Dyn. (USSR), 6 (1977), 863-870. [6] A. V. Borisov, I. S. Mamaev and A. A. Kilin, Absolute and relative choreographies in the problem of point vortices moving on a plane,, Regular and Chaotic Dynamics, 9 (2004), 101-111.  doi: 10.1070/RD2004v009n02ABEH000269. [7] A. V. Borisov, I. S. Mamaev and A. A. Kilin, New periodic solutions for three or four identical vortices on a plane and a sphere, Discrete and Continuous Dynamical Systems - Series B, 2005,110–120. [8] R. Calleja, E. Doedel and C. García-Azpeitia, Symmetries and choreographies in families bifurcating from the polygonal relative equilibrium of the n-body problem,, Celest. Mech. Dyn. Astr., 130 (2018), Art. 48, 28 pp. doi: 10.1007/s10569-018-9841-9. [9] R. Calleja, E. Doedel and C. García-Azpeitia, Choreographies of the$n$-vortex problem,, Regular and Chaotic Dynamics, 23 (2018), 595-612.  doi: 10.1134/S156035471805009X. [10] A. C. Carvalho and H. E. Cabral, Lyapunov Orbits in the n-Vortex Problem,, Regular and Chaotic Dynamics, 19 (2014), 348-362.  doi: 10.1134/S156035471403006X. [11] A. Chenciner and J. Fejoz, Unchained polygons and the n-body problem,, Regular and chaotic dynamics, 14 (2009), 64-115.  doi: 10.1134/S1560354709010079. [12] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses,, Ann. of Math., 152 (2000), 881-901.  doi: 10.2307/2661357. [13] Q. Dai, B. Gebhard and T. Bartsch, Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary,, SIAM Journal on Applied Mathematics, 78 (2018), 977-995.  doi: 10.1137/16M1107085. [14] F. Diacu, Relative Equilibria of the Curved N-Body Problem, Atlantis Series in Dynamical Systems 1. Atlantis Press 2012. doi: 10.2991/978-94-91216-68-8. [15] C. García-Azpeitia and J. Ize, Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators,, J. Differential Equations, 251 (2011), 3202-3227.  doi: 10.1016/j.jde.2011.06.021. [16] C. García-Azpeitia and J. Ize, Bifurcation of periodic solutions from a ring configuration in the vortex and filament problems,, J. Differential Equations, 252 (2012), 5662-5678.  doi: 10.1016/j.jde.2012.01.044. [17] C. García-Azpeitia and J. Ize, Global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the $n$-body problem, J. Differential Equations, 254 (2013), 2033–2075. doi: 10.1016/j.jde.2012.08.022. [18] L. S. Gromeka, On Vortex Motions of Liquid on a Sphere, Collected Papers Moscow, AN USSR, 1952. [19] J. Ize and A. Vignoli, Equivariant Degree Theory, De Gruyter Series in Nonlinear Analysis and Applications 8. Walter de Gruyter, Berlin, 2003. doi: 10.1515/9783110200027. [20] F. Laurent-Polz, J. Montaldi and R. Roberts, Point vortices on the sphere: Stability of symmetric relative equilibria,, J. Geom. Mech., 3 (2011), 439-486.  doi: 10.3934/jgm.2011.3.439. [21] J. Montaldi, R. Roberts and I. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Phil. Trans. R. Soc. Lond. A, 325 (1988), 237-293.  doi: 10.1098/rsta.1988.0053. [22] J. Montaldi and T. Tokieda, Deformation of geometry and bifurcations of vortex rings,, Recent Trends in Dynamical Systems, Springer Basel, 335 (2013), 335–370. doi: 10.1007/978-3-0348-0451-6_14. [23] C. Moore, Braids in classical gravity, Physical Review Letters, 70 (1993), 3675-3679.  doi: 10.1103/PhysRevLett.70.3675. [24] P. K. Newton, The N-vortex Problem, Analytical Techniques, Applied Mathematical Sciences, 145. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3. [25] C. Simó, New families of solutions in N-body problems, European Congress of Mathematics, 101–115, Progr. Math., 201, Birkhäuser, Basel, 2001. [26] J. Vankerschaver and M. Leok, A novel formulation of point vortex dynamics on the sphere: Geometrical and numerical aspects,, Journal of Nonlinear Science, 24 (2013), 1-37.  doi: 10.1007/s00332-013-9182-5.

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##### References:
 [1] Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, 1. American Institute of Mathematical Sciences (AIMS), 2006. [2] T. Bartsch, Topological Methods for Variational Problems with Symmetries, Lecture Notes in Mathematics 1560. Springer-Verlag, 1993. doi: 10.1007/BFb0073859. [3] T. Bartsch and Q. Dai, Periodic solutions of the N-vortex Hamiltonian system in planar domains,, J. Differential Equations, 260 (2016), 2275-2295.  doi: 10.1016/j.jde.2015.10.002. [4] S. Boatto and H. Cabral, Nonlinear stability of a latitudinal ring of point-vortices on a nonrotating sphere,, SIAM J. Appl. Math., 64 (2003), 216-230.  doi: 10.1137/S0036139902399965. [5] V. A. Bogolmonov, Dynamics of vorticity at a sphere, Fluid. Dyn. (USSR), 6 (1977), 863-870. [6] A. V. Borisov, I. S. Mamaev and A. A. Kilin, Absolute and relative choreographies in the problem of point vortices moving on a plane,, Regular and Chaotic Dynamics, 9 (2004), 101-111.  doi: 10.1070/RD2004v009n02ABEH000269. [7] A. V. Borisov, I. S. Mamaev and A. A. Kilin, New periodic solutions for three or four identical vortices on a plane and a sphere, Discrete and Continuous Dynamical Systems - Series B, 2005,110–120. [8] R. Calleja, E. Doedel and C. García-Azpeitia, Symmetries and choreographies in families bifurcating from the polygonal relative equilibrium of the n-body problem,, Celest. Mech. Dyn. Astr., 130 (2018), Art. 48, 28 pp. doi: 10.1007/s10569-018-9841-9. [9] R. Calleja, E. Doedel and C. García-Azpeitia, Choreographies of the$n$-vortex problem,, Regular and Chaotic Dynamics, 23 (2018), 595-612.  doi: 10.1134/S156035471805009X. [10] A. C. Carvalho and H. E. Cabral, Lyapunov Orbits in the n-Vortex Problem,, Regular and Chaotic Dynamics, 19 (2014), 348-362.  doi: 10.1134/S156035471403006X. [11] A. Chenciner and J. Fejoz, Unchained polygons and the n-body problem,, Regular and chaotic dynamics, 14 (2009), 64-115.  doi: 10.1134/S1560354709010079. [12] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses,, Ann. of Math., 152 (2000), 881-901.  doi: 10.2307/2661357. [13] Q. Dai, B. Gebhard and T. Bartsch, Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary,, SIAM Journal on Applied Mathematics, 78 (2018), 977-995.  doi: 10.1137/16M1107085. [14] F. Diacu, Relative Equilibria of the Curved N-Body Problem, Atlantis Series in Dynamical Systems 1. Atlantis Press 2012. doi: 10.2991/978-94-91216-68-8. [15] C. García-Azpeitia and J. Ize, Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators,, J. Differential Equations, 251 (2011), 3202-3227.  doi: 10.1016/j.jde.2011.06.021. [16] C. García-Azpeitia and J. Ize, Bifurcation of periodic solutions from a ring configuration in the vortex and filament problems,, J. Differential Equations, 252 (2012), 5662-5678.  doi: 10.1016/j.jde.2012.01.044. [17] C. García-Azpeitia and J. Ize, Global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the $n$-body problem, J. Differential Equations, 254 (2013), 2033–2075. doi: 10.1016/j.jde.2012.08.022. [18] L. S. Gromeka, On Vortex Motions of Liquid on a Sphere, Collected Papers Moscow, AN USSR, 1952. [19] J. Ize and A. Vignoli, Equivariant Degree Theory, De Gruyter Series in Nonlinear Analysis and Applications 8. Walter de Gruyter, Berlin, 2003. doi: 10.1515/9783110200027. [20] F. Laurent-Polz, J. Montaldi and R. Roberts, Point vortices on the sphere: Stability of symmetric relative equilibria,, J. Geom. Mech., 3 (2011), 439-486.  doi: 10.3934/jgm.2011.3.439. [21] J. Montaldi, R. Roberts and I. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Phil. Trans. R. Soc. Lond. A, 325 (1988), 237-293.  doi: 10.1098/rsta.1988.0053. [22] J. Montaldi and T. Tokieda, Deformation of geometry and bifurcations of vortex rings,, Recent Trends in Dynamical Systems, Springer Basel, 335 (2013), 335–370. doi: 10.1007/978-3-0348-0451-6_14. [23] C. Moore, Braids in classical gravity, Physical Review Letters, 70 (1993), 3675-3679.  doi: 10.1103/PhysRevLett.70.3675. [24] P. K. Newton, The N-vortex Problem, Analytical Techniques, Applied Mathematical Sciences, 145. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3. [25] C. Simó, New families of solutions in N-body problems, European Congress of Mathematics, 101–115, Progr. Math., 201, Birkhäuser, Basel, 2001. [26] J. Vankerschaver and M. Leok, A novel formulation of point vortex dynamics on the sphere: Geometrical and numerical aspects,, Journal of Nonlinear Science, 24 (2013), 1-37.  doi: 10.1007/s00332-013-9182-5.
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