American Institute of Mathematical Sciences

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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

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
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Ph.D Thesis, University of Nice, 2002. Google Scholar

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SIAM J. Math. Anal., 19 (1988), 1295-1313.  Google Scholar

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J. Montaldi and T. Tokieda, A family of point vortex systems,, in preparation., ().   Google Scholar

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Comm. Math. Phys., 69 (1979), 19-30.  Google Scholar

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J. Geom. Phys., 9 (1992), 111-119.  Google Scholar

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Math. Z., 232 (1999), 747-788.  Google Scholar

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J. Fluid Mech., 255 (1993), 35-64.  Google Scholar

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J. Math. Phys., 39 (1998), 5894-5907.  Google Scholar

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show all references

References:
[1]

Adv. Appl. Mech., 39 (2003), 1-79. Google Scholar

[2]

Fluid Dyn., 6 (1977), 863-870. Google Scholar

[3]

SIAM J. Appl. Math., 64 (2003), 216-230.  Google Scholar

[4]

in "Geometric Mechanics and Symmetry," Based on lectures by Montaldi, LMS Lecture Note Series, 306, Cambridge University Press, Cambridge, (2005), 357-402.  Google Scholar

[5]

Regular and Chaotic Dynamics, 8 (2003), 259-282.  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]

Arch. Ration. Mech. Anal., 163 (2002), 1-33. Correction to "Hamiltonian Hopf bifurcation with symmetry'', Arch. Ration. Mech. Anal., 167 (2002), 83-84.  Google Scholar

[8]

Nonlinearity, 15 (2002), 531-549.  Google Scholar

[9]

With an appendix by Jerrold Marsden, Physica D, 24 (1987), 391-405.  Google Scholar

[10]

English translation, On integrals of the hydrodynamical equations which express vortex motion, Phil. Mag., 33 (1867), 485-512. Google Scholar

[11]

Prentice-Hall, 1975. Google Scholar

[12]

Physica D, 116 (1998), 143-175.  Google Scholar

[13]

Kap.\ XX, Teubner, Leipzig, 1876. Google Scholar

[14]

Chaos, 14 (2004), 592-602.  Google Scholar

[15]

Nonlinearity, 15 (2002), 143-171.  Google Scholar

[16]

Nonlinearity, 17 (2004), 1989-2013.  Google Scholar

[17]

Regul. Chaotic Dyn., 10 (2005), 39-58.  Google Scholar

[18]

Ph.D Thesis, University of Nice, 2002. Google Scholar

[19]

Physica D, 148 (2001), 97-135.  Google Scholar

[20]

Nuovo Cimento C, 22 (1999), 793-802. Google Scholar

[21]

Lecture Notes in Mathematics, 1160, Springer-Verlag, Berlin, 1985.  Google Scholar

[22]

Phys. Fluids, 21 (1978), 1092-1095. Google Scholar

[23]

Celest. Mech., 4 (1971), 99-109.  Google Scholar

[24]

SIAM J. Math. Anal., 19 (1988), 1295-1313.  Google Scholar

[25]

Nonlinearity, 10 (1997), 449-466.  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. Nonlinear Sci., 9 (1999), 53-88.  Google Scholar

[29]

Phil. Trans. R. Soc. London Ser. A, 325 (1988), 237-293.  Google Scholar

[30]

Nonlinearity, 3 (1990), 695-730.  Google Scholar

[31]

SIAM J. on Applied Dynamical Systems, 2 (2003), 417-430.  Google Scholar

[32]

J. Montaldi and T. Tokieda, A family of point vortex systems,, in preparation., ().   Google Scholar

[33]

Applied Mathematical Sciences, 145, Springer-Verlag, New York, 2001.  Google Scholar

[34]

Ph.D Thesis, University of California, Santa Cruz, 1998. Google Scholar

[35]

Comm. Math. Phys., 69 (1979), 19-30.  Google Scholar

[36]

J. Geom. Phys., 9 (1992), 111-119.  Google Scholar

[37]

Math. Z., 232 (1999), 747-788.  Google Scholar

[38]

J. Fluid Mech., 255 (1993), 35-64.  Google Scholar

[39]

J. Math. Phys., 39 (1998), 5894-5907.  Google Scholar

[40]

Third revised edition, Hermann, Paris, 1978.  Google Scholar

[41]

J. Fluid Mech., 460 (2002), 83-92.  Google Scholar

[42]

C.R. Acad. Sci. Paris Série I Math., 333 (2001), 943-946.  Google Scholar

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