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

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

Carlos García-Azpeitia

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.

Keywords: $ n $-vortex problem, choreographies, global bifurcation, equivariant degree theory.
Mathematics Subject Classification: 34C25, 37G40, 47H11, 54F45.
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.  Google Scholar

[2]

T. Bartsch, Topological Methods for Variational Problems with Symmetries, Lecture Notes in Mathematics 1560. Springer-Verlag, 1993. doi: 10.1007/BFb0073859.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

V. A. Bogolmonov, Dynamics of vorticity at a sphere, Fluid. Dyn. (USSR), 6 (1977), 863-870.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

L. S. Gromeka, On Vortex Motions of Liquid on a Sphere, Collected Papers Moscow, AN USSR, 1952.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

C. Moore, Braids in classical gravity, Physical Review Letters, 70 (1993), 3675-3679.  doi: 10.1103/PhysRevLett.70.3675.  Google Scholar

[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.  Google Scholar

[25]

C. Simó, New families of solutions in N-body problems, European Congress of Mathematics, 101–115, Progr. Math., 201, Birkhäuser, Basel, 2001.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

T. Bartsch, Topological Methods for Variational Problems with Symmetries, Lecture Notes in Mathematics 1560. Springer-Verlag, 1993. doi: 10.1007/BFb0073859.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

V. A. Bogolmonov, Dynamics of vorticity at a sphere, Fluid. Dyn. (USSR), 6 (1977), 863-870.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

L. S. Gromeka, On Vortex Motions of Liquid on a Sphere, Collected Papers Moscow, AN USSR, 1952.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

C. Moore, Braids in classical gravity, Physical Review Letters, 70 (1993), 3675-3679.  doi: 10.1103/PhysRevLett.70.3675.  Google Scholar

[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.  Google Scholar

[25]

C. Simó, New families of solutions in N-body problems, European Congress of Mathematics, 101–115, Progr. Math., 201, Birkhäuser, Basel, 2001.  Google Scholar

[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.  Google Scholar

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