Bifurcations of Liouville tori in a system of two vortices of positive intensity in a Bose-Einstein condensate

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In this paper we consider a completely Liouville integrable Hamiltonian system with two degrees of freedom, which describes the dynamics of two vortex filaments in a Bose-Einstein condensate enclosed in a harmonic trap. For vortex pairs of positive intensity detected bifurcation of three Liouville tori into one. Such bifurcation was found in the integrable case of Goryachev-Chaplygin-Sretensky in the dynamics of a rigid body. For the integrable perturbation of the physical parameter of the intensity ratio, identified bifurcation proved to be unstable, which led to bifurcations of the type of two tori into one and vice versa.

About the authors

P. E. Ryabov

Finance University under the Government of the Russian Federation; Institute of Machines Sciense named after A.A. Blagonravov of the Russian Academy of Sciences; Udmurt State University

Author for correspondence.

Russian Federation, 49, Leningradsky avenue, Moscow, 125993; 4, Kcharitonievsky, Moscow, 101990; 1/1, Universitetskaya Street, Izhevsk, Udmurtia, 426034


  1. Fetter A. L. Rotating Trapped Bose-Einstein Condensates // Rev. Mod. Phys. 2009. V. 81. № 2. P. 647-691.
  2. Torres P. J., Kevrekidis P. G., Frantzeskakis D. J., Carretero-Gonzalez R., Schmelcher P., Hall D. S. Dyna- mics of Vortex Dipoles in Confined Bose-Einstein Condensates // Phys. Lett. A. 2011. V. 375. P. 3044-3050.
  3. Borisov А. V., Kilin A.A. Stability of Thomson’s Configurations of Vortices on a Sphere // Reg. and Chaot. Dyn. 2000. V. 5. № 2. P. 189-200.
  4. Borisov A. V., Mamaev I. S., Kilin A. A. Absolute and Relative Choreographies in the Problem of Point Vortices Moving on a Plane // Reg. and Chaot. Dyn. 2004. V. 9. № 2. P. 101-111.
  5. Borisov A. V., Kilin A. A., Mamaev I. S. The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem // Reg. and Chaot. Dyn. 2013. V. 18. № 1/2. P. 33-62.
  6. Borisov A. V., Ryabov P. E., Sokolov S. V. Bifurcation Analysis of the Motion of a Cylinder and a Point Vortex in an Ideal Fluid // Math. Notes. 2016. V. 99. № 6. P. 834-839.
  7. Харламов М. П. Топологический анализ интегируемых задач динамики твердого тела. Л.: Изд-во ЛГУ, 1988. 200 c.
  8. Болсинов А. В., Матвеев С. В., Фоменко А. Т. Топологическая классификация интегрируемых гамильтоновых систем с двумя степенями свободы. Список систем малой сложности // УМН. 1990. Т. 45. В. 2 (272). C. 49-77.
  9. Ошемков А. А., Тужилин М. А. Интегрируемые возмущения седловых особенностей ранга 0 гамильтоновых систем // Мат. сб. 2018. Т. 209. № 9. C. 102-127.
  10. Navarro R., Carretero-González R., Torres P. J., Kevrekidis P. G., Frantzeskakis D. J., Ray M. W., Altunta E., Hall D. S. Dynamics of Few Co-Rotating Vortices in Bose-Einstein Condensates // Phys. Rev. Lett. 2013. V. 110. № 22. P. 225301-6.
  11. Sokolov S. V., Ryabov P. E. Bifurcation Analysis of the Dynamics of Two Vortices in a Bose-Einstein Condensate. The Case of Intensities of Opposite Signs // Reg. and Chaot. Dyn. 2017. V. 22. № 8. P. 979-998.
  12. Соколов С.В., Рябов П.Е. // ДАН. 2018. Т. 97. № 6. С. 652-656.
  13. Kharlamov M. P. Extensions of the Appelrot Classes for the Generalized Gyrostat in a Double Force Field // Reg. and Chaot. Dyn. 2014. V. 19. № 2. P. 226-244.
  14. Болсинов А. В., Борисов А. В., Мамаев И. C. Топология и устойчивость интегрируемых систем // УМН. 2010. Т. 65. В. 2. C. 71-132.
  15. Килин А. А., Борисов А. В., Мамаев И. С. Динамика точечных вихрей внутри и вне круговой области. В сб.: Фундаментальные и прикладные проблемы теории вихрей. М.; Ижевск: ИКИ, 2003. 704 с.



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