Point vortices on the sphere: Stability of symmetric relative equilibria

Previous Article
Sobolev metrics on shape space of surfaces
JGM Home
This Issue
Next Article
Covariantizing classical field theories

December 2011, 3(4): 439-486. doi: 10.3934/jgm.2011.3.439

Point vortices on the sphere: Stability of symmetric relative equilibria

Frederic Laurent-Polz ^1, , James Montaldi ^2, and Mark Roberts ^3,

1.	Institut Non Linéaire de Nice, 1361 route des Lucioles, 06560 Valbonne, France
2.	School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom
3.	Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom

Received March 2011 Revised May 2011 Published February 2012

Abstract
Related Papers

We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortices. Such configurations have dihedral symmetry, and the symmetry is used to block diagonalize the relevant matrices, to distinguish the subspaces on which their eigenvalues need to be calculated, and also to describe the bifurcations that occur as eigenvalues pass through zero.

Keywords: bifurcations, Hamiltonian systems, stability, symmetry., Point vortices.

Mathematics Subject Classification: Primary: 37J15, 37J20, 37J25, 70H33, 76B4.

Citation: Frederic Laurent-Polz, James Montaldi, Mark Roberts. Point vortices on the sphere: Stability of symmetric relative equilibria. Journal of Geometric Mechanics, 2011, 3 (4) : 439-486. doi: 10.3934/jgm.2011.3.439

References:

[1]	H. Aref, P. Newton, M. Stremler, T. Tokieda and D. L. Vainchtein, Vortex crystals., Adv. Appl. Mech., 39 (2003), 1. Google Scholar
[2]	V. Bogomolov, Dynamics of vorticity at a sphere,, Fluid Dyn., 6 (1977), 863. Google Scholar
[3]	S. Boatto and H. E. Cabral, Nonlinear stability of a latitudinal ring of point-vortices on a nonrotating sphere,, SIAM J. Appl. Math., 64 (2003), 216. Google Scholar
[4]	P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian Bifurcations,, in, 306 (2005), 357. Google Scholar
[5]	H. E. Cabral, K. R. Meyer and D. S. Schmidt, Stability and bifurcations for the $N+1$ vortex problem on the sphere,, Regular and Chaotic Dynamics, 8 (2003), 259. Google Scholar
[6]	H. E. Cabral and D. S. Schmidt, Stability of relative equilibria in the problem of $N+1$ vortices,, SIAM J. Math. Anal., 31 (): 231. Google Scholar
[7]	P. Chossat, J.-P. Ortega and T. Ratiu, Hamiltonian Hopf bifurcation with symmetry,, Arch. Ration. Mech. Anal., 163 (2002), 1. Google Scholar
[8]	G. Derks and T. Ratiu, Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation,, Nonlinearity, 15 (2002), 531. Google Scholar
[9]	M. Golubitsky and I. Stewart, Generic bifurcation of Hamiltonian systems with symmetry,, With an appendix by Jerrold Marsden, 24 (1987), 391. Google Scholar
[10]	H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen, Crelles J., 55 (1858), 25-55., English translation, 33 (1867), 485. Google Scholar
[11]	E. Hansen, "A Table of Series and Products,'', Prentice-Hall, (1975). Google Scholar
[12]	R. Kidambi and P. Newton, Motion of three point vortices on a sphere,, Physica D, 116 (1998), 143. Google Scholar
[13]	G. Kirchhoff, "Vorlesungen über Mathematische Physik, Mechanik,'', Kap.\ XX, (1876). Google Scholar
[14]	L. G. Kurakin, On the nonlinear stability of the regular vortex systems on a sphere,, Chaos, 14 (2004), 592. Google Scholar
[15]	F. Laurent-Polz, Point vortices on the sphere: A case with opposite vorticities,, Nonlinearity, 15 (2002), 143. Google Scholar
[16]	F. Laurent-Polz, Relative periodic orbits in point vortex systems,, Nonlinearity, 17 (2004), 1989. Google Scholar
[17]	F. Laurent-Polz, Point vortices on a rotating sphere,, Regul. Chaotic Dyn., 10 (2005), 39. Google Scholar
[18]	F. Laurent-Polz, "Etude Géométrique de la Dynamique de $N$ Tourbillons Ponctuels sur une Sphère,'', Ph.D Thesis, (2002). Google Scholar
[19]	C. Lim, J. Montaldi and M. Roberts, Relative equilibria of point vortices on the sphere,, Physica D, 148 (2001), 97. Google Scholar
[20]	J. Marsden, S. Pekarsky and S. Shkoller, Stability of relative equilibria of point vortices on a sphere and symplectic integrators,, Nuovo Cimento C, 22 (1999), 793. Google Scholar
[21]	J.-C. van der Meer, "The Hamiltonian Hopf Bifurcation,'', Lecture Notes in Mathematics, 1160 (1160). Google Scholar
[22]	G. J. Mertz, Stability of body-centered polygonal configurations of ideal vortices,, Phys. Fluids, 21 (1978), 1092. Google Scholar
[23]	K. R. Meyer and D. S. Schmidt, Periodic orbits near L4 for mass ratios near the critical mass ratio of Routh,, Celest. Mech., 4 (1971), 99. Google Scholar
[24]	K. R. Meyer and D. S. Schmidt, Bifurcations of relative equilibria in the $N$-body and Kirchhoff problems,, SIAM J. Math. Anal., 19 (1988), 1295. Google Scholar
[25]	J. Montaldi, Persistence and stability of relative equilibria,, Nonlinearity, 10 (1997), 449. Google Scholar
[26]	J. Montaldi, Bifurcations of relative equilibria near zero momentum in $\SO(3)$-symmetric Hamiltonian systems,, in preparation., (). Google Scholar
[27]	J. Montaldi, Web pages,, \url{http://www.maths.manchester.ac.uk/jm/Vortices}, (). Google Scholar
[28]	J. Montaldi and M. Roberts, Relative equilibria of molecules,, J. Nonlinear Sci., 9 (1999), 53. Google Scholar
[29]	J. Montaldi, M. Roberts and I. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems,, Phil. Trans. R. Soc. London Ser. A, 325 (1988), 237. Google Scholar
[30]	J. Montaldi, M. Roberts and I. Stewart, Existence of nonlinear normal modes of symmetric Hamiltonian systems,, Nonlinearity, 3 (1990), 695. Google Scholar
[31]	J. Montaldi, A. Soulière and T. Tokieda, Vortex dynamics on a cylinder,, SIAM J. on Applied Dynamical Systems, 2 (2003), 417. Google Scholar
[32]	J. Montaldi and T. Tokieda, A family of point vortex systems,, in preparation., (). Google Scholar
[33]	P. Newton, "The $N$-Vortex Problem. Analytical Techniques,'', Applied Mathematical Sciences, 145 (2001). Google Scholar
[34]	J.-P. Ortega, "Symmetry, Reduction, and Stability in Hamiltonian Systems,'', Ph.D Thesis, (1998). Google Scholar
[35]	R. Palais, Principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19. Google Scholar
[36]	G. Patrick, Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space,, J. Geom. Phys., 9 (1992), 111. Google Scholar
[37]	G. Patrick, Dynamics near relative equilibria: Nongeneric momenta at a 1:1 group-reduced resonance,, Math. Z., 232 (1999), 747. Google Scholar
[38]	L. Polvani and D. Dritschel, Wave and vortex dynamics on the surface of a sphere,, J. Fluid Mech., 255 (1993), 35. Google Scholar
[39]	S. Pekarsky and J. Marsden, Point vortices on a sphere: Stability of relative equilibria,, J. Math. Phys., 39 (1998), 5894. Google Scholar
[40]	J.-P. Serre, "Représentations Linéaires des Groupes Finis,'', Third revised edition, (1978). Google Scholar
[41]	A. Soulière and T. Tokieda, Periodic motions of vortices on surfaces with symmetry,, J. Fluid Mech., 460 (2002), 83. Google Scholar
[42]	T. Tokieda, Tourbillons dansants,, C.R. Acad. Sci. Paris Série I Math., 333 (2001), 943. Google Scholar

show all references

References:

[1]	H. Aref, P. Newton, M. Stremler, T. Tokieda and D. L. Vainchtein, Vortex crystals., Adv. Appl. Mech., 39 (2003), 1. Google Scholar
[2]	V. Bogomolov, Dynamics of vorticity at a sphere,, Fluid Dyn., 6 (1977), 863. Google Scholar
[3]	S. Boatto and H. E. Cabral, Nonlinear stability of a latitudinal ring of point-vortices on a nonrotating sphere,, SIAM J. Appl. Math., 64 (2003), 216. Google Scholar
[4]	P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian Bifurcations,, in, 306 (2005), 357. Google Scholar
[5]	H. E. Cabral, K. R. Meyer and D. S. Schmidt, Stability and bifurcations for the $N+1$ vortex problem on the sphere,, Regular and Chaotic Dynamics, 8 (2003), 259. Google Scholar
[6]	H. E. Cabral and D. S. Schmidt, Stability of relative equilibria in the problem of $N+1$ vortices,, SIAM J. Math. Anal., 31 (): 231. Google Scholar
[7]	P. Chossat, J.-P. Ortega and T. Ratiu, Hamiltonian Hopf bifurcation with symmetry,, Arch. Ration. Mech. Anal., 163 (2002), 1. Google Scholar
[8]	G. Derks and T. Ratiu, Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation,, Nonlinearity, 15 (2002), 531. Google Scholar
[9]	M. Golubitsky and I. Stewart, Generic bifurcation of Hamiltonian systems with symmetry,, With an appendix by Jerrold Marsden, 24 (1987), 391. Google Scholar
[10]	H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen, Crelles J., 55 (1858), 25-55., English translation, 33 (1867), 485. Google Scholar
[11]	E. Hansen, "A Table of Series and Products,'', Prentice-Hall, (1975). Google Scholar
[12]	R. Kidambi and P. Newton, Motion of three point vortices on a sphere,, Physica D, 116 (1998), 143. Google Scholar
[13]	G. Kirchhoff, "Vorlesungen über Mathematische Physik, Mechanik,'', Kap.\ XX, (1876). Google Scholar
[14]	L. G. Kurakin, On the nonlinear stability of the regular vortex systems on a sphere,, Chaos, 14 (2004), 592. Google Scholar
[15]	F. Laurent-Polz, Point vortices on the sphere: A case with opposite vorticities,, Nonlinearity, 15 (2002), 143. Google Scholar
[16]	F. Laurent-Polz, Relative periodic orbits in point vortex systems,, Nonlinearity, 17 (2004), 1989. Google Scholar
[17]	F. Laurent-Polz, Point vortices on a rotating sphere,, Regul. Chaotic Dyn., 10 (2005), 39. Google Scholar
[18]	F. Laurent-Polz, "Etude Géométrique de la Dynamique de $N$ Tourbillons Ponctuels sur une Sphère,'', Ph.D Thesis, (2002). Google Scholar
[19]	C. Lim, J. Montaldi and M. Roberts, Relative equilibria of point vortices on the sphere,, Physica D, 148 (2001), 97. Google Scholar
[20]	J. Marsden, S. Pekarsky and S. Shkoller, Stability of relative equilibria of point vortices on a sphere and symplectic integrators,, Nuovo Cimento C, 22 (1999), 793. Google Scholar
[21]	J.-C. van der Meer, "The Hamiltonian Hopf Bifurcation,'', Lecture Notes in Mathematics, 1160 (1160). Google Scholar
[22]	G. J. Mertz, Stability of body-centered polygonal configurations of ideal vortices,, Phys. Fluids, 21 (1978), 1092. Google Scholar
[23]	K. R. Meyer and D. S. Schmidt, Periodic orbits near L4 for mass ratios near the critical mass ratio of Routh,, Celest. Mech., 4 (1971), 99. Google Scholar
[24]	K. R. Meyer and D. S. Schmidt, Bifurcations of relative equilibria in the $N$-body and Kirchhoff problems,, SIAM J. Math. Anal., 19 (1988), 1295. Google Scholar
[25]	J. Montaldi, Persistence and stability of relative equilibria,, Nonlinearity, 10 (1997), 449. Google Scholar
[26]	J. Montaldi, Bifurcations of relative equilibria near zero momentum in $\SO(3)$-symmetric Hamiltonian systems,, in preparation., (). Google Scholar
[27]	J. Montaldi, Web pages,, \url{http://www.maths.manchester.ac.uk/jm/Vortices}, (). Google Scholar
[28]	J. Montaldi and M. Roberts, Relative equilibria of molecules,, J. Nonlinear Sci., 9 (1999), 53. Google Scholar
[29]	J. Montaldi, M. Roberts and I. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems,, Phil. Trans. R. Soc. London Ser. A, 325 (1988), 237. Google Scholar
[30]	J. Montaldi, M. Roberts and I. Stewart, Existence of nonlinear normal modes of symmetric Hamiltonian systems,, Nonlinearity, 3 (1990), 695. Google Scholar
[31]	J. Montaldi, A. Soulière and T. Tokieda, Vortex dynamics on a cylinder,, SIAM J. on Applied Dynamical Systems, 2 (2003), 417. Google Scholar
[32]	J. Montaldi and T. Tokieda, A family of point vortex systems,, in preparation., (). Google Scholar
[33]	P. Newton, "The $N$-Vortex Problem. Analytical Techniques,'', Applied Mathematical Sciences, 145 (2001). Google Scholar
[34]	J.-P. Ortega, "Symmetry, Reduction, and Stability in Hamiltonian Systems,'', Ph.D Thesis, (1998). Google Scholar
[35]	R. Palais, Principle of symmetric criticality,, Comm. Math. Phys., 69 (1979), 19. Google Scholar
[36]	G. Patrick, Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space,, J. Geom. Phys., 9 (1992), 111. Google Scholar
[37]	G. Patrick, Dynamics near relative equilibria: Nongeneric momenta at a 1:1 group-reduced resonance,, Math. Z., 232 (1999), 747. Google Scholar
[38]	L. Polvani and D. Dritschel, Wave and vortex dynamics on the surface of a sphere,, J. Fluid Mech., 255 (1993), 35. Google Scholar
[39]	S. Pekarsky and J. Marsden, Point vortices on a sphere: Stability of relative equilibria,, J. Math. Phys., 39 (1998), 5894. Google Scholar
[40]	J.-P. Serre, "Représentations Linéaires des Groupes Finis,'', Third revised edition, (1978). Google Scholar
[41]	A. Soulière and T. Tokieda, Periodic motions of vortices on surfaces with symmetry,, J. Fluid Mech., 460 (2002), 83. Google Scholar
[42]	T. Tokieda, Tourbillons dansants,, C.R. Acad. Sci. Paris Série I Math., 333 (2001), 943. Google Scholar

[1]	James Montaldi. Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry. Journal of Geometric Mechanics, 2014, 6 (2) : 237-260. doi: 10.3934/jgm.2014.6.237
[2]	Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
[3]	Takashi Suzuki. Brownian point vortices and dd-model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 161-176. doi: 10.3934/dcdss.2014.7.161
[4]	A.V. Borisov, A.A. Kilin, I.S. Mamaev. Reduction and chaotic behavior of point vortices on a plane and a sphere. Conference Publications, 2005, 2005 (Special) : 100-109. doi: 10.3934/proc.2005.2005.100
[5]	Joris Vankerschaver, Eva Kanso, Jerrold E. Marsden. The geometry and dynamics of interacting rigid bodies and point vortices. Journal of Geometric Mechanics, 2009, 1 (2) : 223-266. doi: 10.3934/jgm.2009.1.223
[6]	A. V. Borisov, I. S. Mamaev, S. M. Ramodanov. Dynamics of a circular cylinder interacting with point vortices. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 35-50. doi: 10.3934/dcdsb.2005.5.35
[7]	Stefanella Boatto. Curvature perturbations and stability of a ring of vortices. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 349-375. doi: 10.3934/dcdsb.2008.10.349
[8]	Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207
[9]	Mark A. Pinsky, Alexandr A. Zevin. Stability criteria for linear Hamiltonian systems with uncertain bounded periodic coefficients. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 243-250. doi: 10.3934/dcds.2005.12.243
[10]	Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 734-741. doi: 10.3934/proc.2003.2003.734
[11]	Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67
[12]	Luca Biasco, Luigi Chierchia. On the stability of some properly--degenerate Hamiltonian systems with two degrees of freedom. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 233-262. doi: 10.3934/dcds.2003.9.233
[13]	Kristoffer Varholm. Solitary gravity-capillary water waves with point vortices. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3927-3959. doi: 10.3934/dcds.2016.36.3927
[14]	Colm Connaughton, John R. Ockendon. Interactions of point vortices in the Zabusky-McWilliams model with a background flow. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1795-1807. doi: 10.3934/dcdsb.2012.17.1795
[15]	Tiphaine Jézéquel, Patrick Bernard, Eric Lombardi. Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153
[16]	Jianhe Shen, Shuhui Chen, Kechang Lin. Study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 231-254. doi: 10.3934/dcdsb.2011.15.231
[17]	Anatoly Neishtadt. On stability loss delay for dynamical bifurcations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 897-909. doi: 10.3934/dcdss.2009.2.897
[18]	Thierry Gallay. Stability and interaction of vortices in two-dimensional viscous flows. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1091-1131. doi: 10.3934/dcdss.2012.5.1091
[19]	Anna Goƚȩbiewska, Norimichi Hirano, Sƚawomir Rybicki. Global symmetry-breaking bifurcations of critical orbits of invariant functionals. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2005-2017. doi: 10.3934/dcdss.2019129
[20]	Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201

2018 Impact Factor: 0.525

American Institute of Mathematical Sciences