On the stability of Thomson’s vortex pentagon inside a circular domain. (English) Zbl 1270.76018

Kurakin, Leonid G.

Regul. Chaotic Dyn. 17, No. 2, 150-169 (2012).

Summary: We investigate the stability problem for stationary rotation of five identical point vortices located at the vertices of a regular pentagon inside a circular domain. The main results are proofs of theorems which have been announced by the author [“Stability, resonances, and instability of regular vortex polygons in a circular domain”, Doklady Physics 49, No. 11, 658–661 (2004)] previously.

MSC:

76B47	Vortex flows for incompressible inviscid fluids
34D20	Stability of solutions to ordinary differential equations
70K30	Nonlinear resonances for nonlinear problems in mechanics

Keywords:

point vortices; stationary motion; stability; resonance

References:

[1]	Meleshko, V.V. and Konstantinov, M.Yu., Dynamics of Vortex Structures, Kiev: Naukova Dumka, 1993 (Russian).
[2]	Newton, P.K., The N-vortex Problem: Analytical Techniques, Appl. Math. Sci., vol. 145, New York: Springer, 2001. · Zbl 0981.76002
[3]	Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., and Vainchtein, D. L., Vortex Crystals, Adv. Appl. Mech., 2003, vol. 39, pp. 1–79.
[4]	Borisov, A.V. and Mamaev, I. S., Mathematical Methods in the Dynamics of Vortex Structures, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).
[5]	Kozlov, V.V., General Theory of Vortices, Encyclopaedia Math. Sci., vol. 67, Berlin-Heidelberg: Springer, 2003. · Zbl 1104.37050
[6]	Kurakin, L. G., On the Stability of the Regular N-sided Polygon of Vortices, Dokl. Akad. Nauk, 1994, vol. 335, no. 6, pp. 729–731 [Dokl. Phys., 1994, vol. 39, no. 4, pp. 284–286]. · Zbl 0832.76028
[7]	Kurakin, L. G. and Yudovich, V. I., The Nonlinear Stability of Steady Rotation of a Regular Vortex Polygon, Dokl. Akad. Nauk, 2002, vol. 384, no. 4, pp. 476–482 [Dokl. Phys., 2002, vol. 47, no. 6, pp. 465–470]. · Zbl 1080.76520
[8]	Kurakin, L. G. and Yudovich, V. I., The Stability of Stationary Rotation of a Regular Vortex Polygon, Chaos, 2002, vol. 12, no. 3, pp. 574–595. · Zbl 1080.76520
[9]	Kurakin, L. G., On the Nonlinear Stability of Regular Vortex Polygons and Polyhedra on a Sphere, Dokl. Akad. Nauk, 2003, vol. 388, no. 4, pp. 482–487 [Dokl. Phys., 2003, vol. 48, no. 2, pp. 84–89]
[10]	Kurakin, L. G., On Nonlinear Stability of the Regular Vortex Systems on a Sphere, Chaos, 2004, vol. 14, no. 3, pp. 592–602. · Zbl 1080.76031
[11]	Kurakin, L. G., Stability, Resonances, and Instability of the Regular Vortex Polygons in the Circular Domain, Dokl. Akad. Nauk, 2004, vol. 399, no. 1, pp. 52–55 [Dokl. Phys., 2004, vol. 49, no. 11, pp. 658–661].
[12]	Havelock, T.H., The Stability of Motion of Rectilinear Vortices in Ring Formation, Philos. Mag., 1931, vol. 11, pp. 617–633. · Zbl 0001.08102
[13]	Kurakin, L. G., On Stability of a Regular Vortex Polygon in the Circular Domain, J. Math. Fluid Mech., 2005, vol. 7, Suppl. 3, pp. S376–S386. · Zbl 1097.76031
[14]	Kurakin, L. G., The Stability of Thomson’s Configurations of Vortices in a Circular Domain, Nelin. Dinam., 2009, vol. 5, no. 3, pp. 295–317 (Russian).
[15]	Kurakin, L. G., On the Stability of Thomson’s Vortex Configurations inside a Circular Domain, Regul. Chaotic Dyn., 2010, vol. 15, no. 1, pp. 40–58. · Zbl 1229.37055
[16]	Milne-Thomson, L.M., Theoretical Hydrodynamics, London: Macmillan, 1968.
[17]	Routh, E. J., A Treatise on the Stability of a Given State Motion, London: Macmillan, 1877.
[18]	Kurakin, L. G. and Yudovich, V. I., The Stability of Stationary Rotation of a Regular Vortex Polygon, in Fundamental and Applied Problems in Vortex Theory, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2003, pp. 238–303 (Russian).
[19]	Proskuryakov, I.V., Problems in Linear Algebra, 3rd ed., Moscow: Nauka, 1966 [Moscow: Mir, 1978; London: Central Books, 1979]. · Zbl 0161.28601
[20]	Kurakin, L. G. and Ostrovskaya, I.V., Stability of the Thomson Vortex Polygon with Evenly Many Vortices outside a Circular Domain, Sibirsk. Mat. Zh., 2010, vol. 51, no. 3, pp. 584–598 [Siberian Math. J., 2010, vol. 51, no. 3, pp. 463–474]. · Zbl 1270.60010
[21]	Kurakin, L. G., The Stability of the Steady Rotation of a System of Three Equidistant Vortices outside a Circle, Prikl. Mat. Mekh., 2011, vol. 75, no. 2, pp. 327–337 [J. Appl. Math. Mech., 2011, vol. 75, no. 2, pp. 227–234]. · Zbl 1272.76116
[22]	Campbell, L. J., Transverse Normal Modes of Finite Vortex Arrays, Phys. Rev. A, 1981, vol. 24, pp. 514–534.
[23]	Markeev, A.P., Libration Points in Celestial Mechanics and Space Dynamics, Moscow: Nauka, 1978 (Russian).
[24]	Sokol’sky, A.G., On Stability of an Autonomous Hamiltonian System with Two Degrees of Freedom under First-order Resonance, Prikl. Mat. Mech., 1977, vol. 41, no. 1, pp. 24–33 [J. Appl. Math. Mech., 1977, vol. 41, no. 1, pp. 20–28].
[25]	Sokol’sky, A.G., On the Stability of an Autonomous Hamiltonian System with Two Degrees of Freedom in the Case of Equal Frequencies, Prikl. Mat. Mech., 1974, vol. 38, no. 5, pp. 791–799 [J. Appl. Math. Mech., 1974, vol. 38, no. 5, pp. 741–749].
[26]	Kovalev, A. M. and Chudnenko, A. N., On the Stability of Equilibrium Positions of a Two-dimensional Hamiltonian System in the Case of Equal Frequencies, Dokl. Akad. Nauk Ukrain. SSR Ser. A, 1977, no. 11, pp. 1011–1014.
[27]	Lerman, L.M. and Markova, A.P., On Stability at the Hamiltonian Hopf Bifurcation, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 148–162. · Zbl 1229.37056
[28]	Kunitsyn, A.N. and Markeev, A. P., Stability in Resonance Cases, in Surveys in Science and Engineering. General Mechanics Series: Vol. 4, Moscow: VINITI, 1979, pp. 58–139. · Zbl 0423.70002
[29]	Markeev, A. P., Stability of a Canonical System with Two Degrees of Freedom in the Presence of Resonance, Prikl. Mat. Mech., 1968, vol. 32, no. 4, pp. 738–744 [J. Appl. Math. Mech., 1968, vol. 32, no. 4, pp. 766–772]. · Zbl 0184.11904
[30]	Arnol’d, V.I., Proof of a Theorem of A.N.Kolmogorov on the Invariance of Quasi-periodic Motions under Small Perturbations of the Hamiltonian, Uspekhi Mat. Nauk, 1963, vol. 18, no. 5, pp. 13–40 [Russian Math. Surveys, 1963, vol. 18, no. 5, pp. 9–36].
[31]	Khazin, L.G. and Shnol’, E.E., Stability of Critical Equilibrium States, Pushchino: Akad. Nauk SSSR, 1985 [Manchester: Manchester Univ. Press, 1985]. · Zbl 0960.34510
[32]	Bryuno, A.D., On Formal Stability of Hamiltonian Systems, Mat. Zametki, 1967, vol. 1, no. 3, pp. 325–330 [Math. Notes, 1967, vol. 1, no. 3, pp. 216–219]. · Zbl 0154.34401
[33]	Arnol’d, V. I. and Avez, A., Ergodic Problems of Classical Mechanics, New York: Benjamin, 1968.
[34]	Moser, J., New Aspects in the Theory of Stability of Hamiltonian Systems, Comm. Pure Appl. Math., 1958, vol. 11, no. 1, pp. 81–114. · Zbl 0082.40801
[35]	Sokol’sky, A.G., Stability of the Lagrange Solutions of the Restricted Three-body Problem for the Critical Ratio of the Masses, Prikl. Mat. Mech., 1975, vol. 39, no. 2, pp. 366–369 [J. Appl. Math. Mech., 1975, vol. 39, no. 2, pp. 342–345].
[36]	Lyapunov, A. M., A Study of One of the Special Cases of the Problem of Stability of Motion, Mat. Sb., 1893, vol. 17, no. 2, pp. 253–333 (Russian); see also: Lyapunov, A. M., The General Problem of the Stability of Motion, Moscow: Gostekhizdat, 1950, pp. 369–450 (Russian); Lyapunov, A. M., The Stability of Motion in a Particular Case of the Three-body Problem, in Collected Works: Vol. 1, Moscow: Akad. Nauk SSSR, 1954, pp. 327–401 (Russian).
[37]	Kurakin, L. G., On the Lyapunov Chain of Stability Criteria in the Critical Case of a Jordan 2-Block, Dokl. Russian Akad. Nauk, 1994, vol. 337, no. 1, pp. 14–16 [Russian Acad. Sci. Dokl. Math., 1995, vol. 50, no. 1, pp. 10–13].
[38]	Kurakin, L. G., On the Stability Criteria in A.M. Lyapunov’s Paper ”A Study of One of the Special Cases of the Problem of Stability of Motion”, Vladikavkaz. Mat. Zh., 2009, vol. 11, no. 3, pp. 28–37 (Russian). · Zbl 1324.34109
[39]	Bryuno, A.D. and Petrov, A.G., On the Calculation of Hamiltonian Normal Form, Dokl. Akad. Nauk, 2006, vol. 410, no. 4, pp. 474–478 [Dokl. Phys., 2006, vol. 51, no. 10, pp. 555–559].
[40]	Bruno, A.D., Power Geometry in Algebraic and Differential Equations, Moscow: Fizmatlit, 1998 [North-Holland Mathematical Library, vol. 57, Amsterdam: Elsevier, 2000].