Abstract
This paper is concerned with the study of the topological type of the level sets of the integrable cases of Armbruster Guckenheimer Kim galactic potential. Furthermore, all generic bifurcation of the level sets are presented. We determine the families of periodic solutions by giving the solution in terms of Jacobi’s elliptic functions. Finally, the phase portrait is studied.
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References
Acosta-Humánez, P.: Galoisian Approach to Supersymmetric Quantum Mechanics the Integrability Analysis of the Schrödinger Equation by Means of Differential Galois Theory. VDM Verlag Dr. Muller, Berlin (2010)
Acosta-Humánez, P., Alvarez-Ramirez, M., Stuchi, T.J.: Nonintegrability of the Armbruster Guckenheimer Kim quartic Hamiltonian through Morales Ramis theory. SIAM J. Appl. Dyn. Syst. 17, 78–96 (2017)
Armbruster, D., Guckenheimer, J., Kim, S.: Chaotic dynamics in systems with square symmetry. Phys. Lett. A 140, 416–420 (1989)
Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, Dynamical System III. Encyclopaedia of Mathematical Science, vol. 3 (2006)
Aronlod, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics. Springer, Berlin (1989)
Bertola, F., Capaccioli, M.: Dynamics of early type galaxies I: the rotation curve of the elliptical galaxy NGC 4697. Astrophys. J. 200, 439–445 (1975)
Caranicolas, N.D.: From global dynamical models to the Hénon-Heiles potential. Mech. Res. Commun. 29, 291–298 (2002)
Contopoulos, G.: On the relative motions of stars in a galaxy. Stockh. Obs. Ann. 19, 10 (1957)
Contopoulos, G.: A third integral of motion in a galaxy. Z. Astrophys. 49, 273–291 (1960)
Contopoulos, G.: On the existence of a third integral of motion. Astron. J. 68, 1–14 (1963)
Dekkaki, S., Lassas, A., Quazzani, A., Quazzani-Jamil, M.: Bifurcations sets of the Sretensky axial symmetric gyrostat. Moroc. J. Condens. Matter 5(1), 52–61 (2004)
Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry—Methods and Applications, Part II: The Geometry and Topology of Manifolds. Springer, New York (1985)
Eddington, A.S.: The dynamics of a stellar system third paper oblate and other distributions. Mon. Not. R. Astron. Soc. 76, 37–60 (1915)
El-Sabaa, F.M.: Bifurcation of Kovalevskaya polynomial. Int. J. Theor. Phys. 10, 2071–2083 (1995)
El-Sabaa, F.M., Hosny, M., Zakria, S.K.: Bifurcations of Liouville tori of a two fixed center problem. Astrophys. Space Sci. 363, 1–10 (2018)
Elmandouh, A.A.: On the dynamics of Armbruster Guckenheimer Kim galactic potential in a rotating reference frame. Astrophys. Space Sci. 361, 182–194 (2016)
Fomenko, A.T.: Integrability and Nonintegrability in Geometry and Mechanics. Kluwer Academic, Norwell (1988)
Fomenko, A.T.: Visual Geometry and Topology. Springer, Berlin (1994)
Gavrilov, L., Ouazzani-Jamil, M., Caboz, R.: Bifurcation diagrams and Fomenko’s surgery on Liouville tori of the Kolossoff potential \(U=\rho +\frac{1}{\rho } -k \cos \phi \). Ann. Sci. Éc. Norm. Supér. 26, 545–564 (1993)
Hori, G.: The motion of a star in the Galaxy. Bull. Astron. Soc. Jpn. 14, 125–127 (1962)
Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73–79 (1964)
Jeans, J.H.: Problems of Cosmogony and Stellar Dynamics p. 233. Cambridge University Press, New York (1919)
Kharbach, J., Dekkaki, S., Quazzani, A., Quazzani-Jamil, M.: Bifurcations of the common level sets of atomic hydrogen in van der Waals potential. Int. J. Bifurc. Chaos 13, 107–114 (2003)
Llibre, J., Roberto, L.: Periodic orbits and non-integrability of Armbruster-Guckenheimer-Kim potential. Astrophys. Space Sci. 343, 69–74 (2012)
Llibre, J., Vidal, C.: Periodic orbits and non-integrability in a cosmological scalar field. J. Math. Phys. 53, 1–16 (2012)
Llibre, J., Vidal, C.: New periodic solution in 3-dimensional galactic-type Hamiltonian systems. Nonlinear Dyn. 78, 968–980 (2014)
Llibre, J., Pasca, D., Valls, C.: Periodic solutions of a galactic potential. Chaos Solitons Fractals 61, 38–43 (2014)
Morales-Ruiz, J.: Differential Galois Theory and Non-integrability of Hamiltonian Systems. Progress in Math. Birkhauser, Basel (1999)
Ollongren, A.: Three-dimensional galactic stellar orbits. Annu. Rev. Astron. Astrophys. 18, 5–63 (1962)
Quazzani, T., Quazzani-Jamil, M.: Bifurcations of Liouville tori of an integrable case of swinging Atwood’s machine. Il Nuovo Cimento B 110, 1111–1121 (1995)
Quazzani, T., Dekkaki, S., Kharbach, J., Quazzani-Jamil, M.: Bifurcation sets of the motion of a heavy rigid body around a fixed point in Goryatchev-Tchaplygin case. Nuovo Cimento 32, 1175–1193 (2000)
Schmidt, M.: A model of the distribution of mass in the Galactic System. Bull. Astron. Inst. Neth. 13, 15–41 (1956)
Vozmischeva, T.G.: Integrable Problems of Celestial Mechanics in Spaces of Constant Curvature. Astrophysics and Space Science Library, Berlin, p. 180 (2003)
Vozmischeva, T.G., Oshemkov, A.A.: The topological analysis of the two-center problem on the two-dimensional sphere. Sb. Math. 193(8), 3–38 (2002)
Zeeuw, T., Merritt, D.: Stellar orbits in a triaxial galaxy I. Orbits in the plane of rotation. Astrophys. J. 267, 571–595 (1983)
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El-Sabaa, F.M., Hosny, M. & Zakria, S.K. Bifurcations of Armbruster Guckenheimer Kim galactic potential. Astrophys Space Sci 364, 34 (2019). https://doi.org/10.1007/s10509-019-3519-y
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DOI: https://doi.org/10.1007/s10509-019-3519-y