Abstract
We investigate the dynamics in a galactic potential with two reflection symmetries. The phase-space structure of the real system is approximated with a resonant detuned normal form constructed with the method based on the Lie transform. Attention is focused on the stability properties of the axial periodic orbits that play an important role in galactic models. Using energy and ellipticity as parameters, we find analytical expressions of bifurcations and compare them with numerical results available in the literature.
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Belmonte, C., Boccaletti, D., Pucacco, G.: Approximate first integrals for a model of galactic potential with the method of Lie transform normalization. In: Perez-Chavela, E., Xia, J. (eds.) Submitted to the Proceedings of the Saarifest (2005)
Birkhoff, G.D.: Dynamical systems, Amer. Math. Soc. Coll. Publ., vol. 9, New York, USA (1927)
Binney, J., Tremaine, S.: Galactic Dynamics, Princeton University Press (1987)
Boccaletti, D., Pucacco, G.: Theory of Orbits, vol. 2, Springer-Verlag, Berlin (1999)
Contopoulos, G.: Higher order resonances in dynamical systems. Cel. Mech. 18, 195–204 (1978)
Contopoulos, G.: Order and Chaos in Dynamical Astronomy. Springer-Verlag, Berlin (2002)
Contopoulos, G., Efthymiopoulos, C., Giorgilli, A.: Nonconvergence of formal integrals of motion. J. Phys. A: Math. Gen. 36, 8639–8660 (2003)
de Zeeuw, T., Merritt, D.: Stellar orbits in a triaxial galaxy. I. Orbits in the plane of rotation. Astrophys. J. 267, 571–595 (1983)
Dragt, A., Finn, J.M.: Lie series and invariant functions for analytic symplectic maps. J. Mat. Phys. 17, 2215–2227 (1976)
Efthymiopoulos, C., Giorgilli, A., Contopoulos, G.: Nonconvergence of formal integrals: II. Improved estimates for the optimal order of truncation. J. Phys. A: Math. Gen. 37, 10831–10858 (2004)
Finn, J.M.: Lie series: a perspective. Local and global methods of nonlinear dynamics. Lecture Notes in Physics, vol. 252, pp. 63–86 (1984)
Fridman, T., Merritt, D.: Periodic orbits in triaxial galaxies with weak cusps. Astron. J. 114, 1479–1487 (1997)
Gustavson, F.: On constructing formal integrals of a Hamiltonian system near an equilibrium point. Astron. J. 71, 670–686 (1966)
Koseleff, P.V.: Comparison between Deprit and Dragt-Finn perturbation methods. Cel. Mech. & Dynam. Astron. 58, 17–36 (1994)
Kummer, M.: On resonant Hamiltonians with two degrees of freedom near an equilibrium point. Lecture Notes in Physics, vol. 93, pp. 57–75 (1977)
Miralda-Escudé, J., Schwarzschild, M.: On the orbit structure of the logarithmic potential. Astrophys. J. 339, 752–762 (1989)
Moser, J.: Lectures on Hamiltonian systems. Mem. Am. Math. Soc. 81, 1–60 (1968)
Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, New York (1985)
Schwarzschild, M.: A numerical model for a triaxial stellar system in dynamical equilibrium. Astrophys. J. 232, 236–247 (1979)
Scuflaire, R.: Stability of axial orbits in analytic galactic potentials. Cel. Mech. & Dynam. Astron. 61, 261–285 (1995)
Tuwankotta, J.M., Verhulst, F.: Symmetry and resonance in Hamiltonian systems. SIAM J. Appl. Math. 61, 1369–1385 (2000)
Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems. Springer-Verlag, Berlin (1996)
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Belmonte, C., Boccaletti, D., Pucacco, G. (2006). Stability of axial orbits in galactic potentials. In: Celletti, A., Ferraz-Mello, S. (eds) Periodic, Quasi-Periodic and Chaotic Motions in Celestial Mechanics: Theory and Applications. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5325-2_6
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DOI: https://doi.org/10.1007/978-1-4020-5325-2_6
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