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Periodic orbits of a perturbed 3-dimensional isotropic oscillator with axial symmetry

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Abstract

We study the periodic orbits of a generalized Yang–Mills Hamiltonian \(\mathcal {H}\) depending on a parameter \(\beta \). Playing with the parameter \(\beta \) we are considering extensions of the Contopoulos and of the Yang–Mills Hamiltonians in a 3-dimensional space. This Hamiltonian consists of a 3-dimensional isotropic harmonic oscillator plus a homogeneous potential of fourth degree having an axial symmetry, which implies that the third component N of the angular momentum is constant. We prove that in each invariant space \(\mathcal {H}=h>0\) the Hamiltonian system has at least four periodic solutions if either \(\beta <0\), or \(\beta = 5+\sqrt{13}\); and at least 12 periodic solutions if \(\beta >6\) and \(\beta \ne 5+\sqrt{13}\). We also study the linear stability or instability of these periodic solutions.

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References

  1. Almeida, M., Moreira, I., Santos, F.: On the Ziglin-Yoshida analysis for some classes of homogeneous Hamiltonian systems. Braz. J. Phys. (1998). doi:10.1590/S0103-97331998000400022

    Google Scholar 

  2. Biswas, D., Azam, M., Lawande, Q., Lawande, S.: Existence of stable periodic orbits in the \(x^2 y^2\) potential: a semiclassical approach. J. Phys. A Math. Gen. 25, 297–301 (1992)

    Article  Google Scholar 

  3. Caranicolas, N., Varvoglis, H.: Families of periodic orbits in a quartic potential. Astron. Astrophys. 141, 383–388 (1984)

    MATH  Google Scholar 

  4. Coffey, S.L., Deprit, A., Miller, B.R.: The critical inclination in artificial satellite theory. Celest. Mech. 39, 365–406 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. Contopoulos, G., Papadaki, H., Polymilis, C.: The structure of chaos in a potential without escapes. Celest. Mech. Dyn. Astron. 60, 249–271 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Contopoulos, G., Harssoula, M., Voglis, N., Dvorak, R.: Destruction of islands of stability. J. Phys. A Math. Gen. 32, 5213–5232 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Contopoulos, G., Efthymiopoulos, C., Giorgilli, A.: Non-convergence of formal integrals of motion. J. Phys. A Math. Gen. 36, 8639–8660 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deprit, A.: The Lissajous transformation I. Basic. Dyn. Astron. Celest. Mech. 51, 201–225 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Elipe, A., Hietarinta, J., Tompadis, S.: Comment on a paper by Kasperczuk. Cel. Mech. 58, 378–391 (1994)

    Google Scholar 

  10. Elipe, A., Hietarinta, J., Tompadis, S.: Comment on a paper by Kasperczuk. Celest. Mech. Dyn. Astron. 62, 191–192 (1995)

    Article  Google Scholar 

  11. Falconi, M., Lacomba, E., Vidal, C.: On the dynamics of mechanical systems with homogeneous polynomial potentials of degree 4. Bull. Braz. Math. Soc. New Ser. 38, 301–333 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ferrer, S., Hanßmann, H., Palacián, J., Yanguas, P.: On perturbed oscillators in 1–1–1 resonance: the case of axially symmetric cubic potentials. J. Geom. Phys. 40, 320–369 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guirao, J.L.G., Llibre, J., Vera, J.A.: Periodic orbits of Hamiltonian systems: applications to perturbed Kepler problems. Chaos Solitons Fractals 57, 105–111 (2013)

    Article  MathSciNet  Google Scholar 

  14. Guirao, J.L.G., Llibre, J., Vera, J.A.: Periodic solutions induced by an upright one for a sleeping symmetrical gyrostat. Nonlinear Dyn. 73, 417–425 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guirao, J.L.G., Llibre, J., Vera, J.A.: On the dynamics of the rigid body with a fixed point: periodic orbits and integrability. Nonlinear Dyn. 74, 327–333 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jiménez-Lara, L., Llibre, J.: Periodic orbits and non-integrability of generalized classical Yang–Mills Hamiltonian system. J. Math. Phys. 52, 032901–032909 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kasperczuc, S.: Integrability of the Yang–Mills Hamiltonian System. Celest. Mech. Dyn. Astron. 58, 387–391 (1994)

    Article  MathSciNet  Google Scholar 

  18. Llibre, J., Ramirez, R., Sadovskaia, N.: Integrability of the constrained rigid body. Nonlinear Dyn. 73, 2273–2290 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Maciejewski, A., Radzki, W., Rybicki, S.: Periodic trajectories near degenerate equilibria in the Henon–Heiles and Yang–Mills Hamiltonian systems. J. Dyn. Syst. Differ. Equ. 17, 475–488 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems, Second edition, Applied Mathematical Sciences. Springer, New York (2007)

    MATH  Google Scholar 

  21. Verhulst, F.: Nonlinear Dierential Equations and Dynamical Systems. Universitext, Springer (1991)

    Google Scholar 

  22. Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge Mathematical Library, Cambridge (1989)

    Google Scholar 

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Acknowledgments

We thank to the reviewers their comments and suggestions which help us to improve the presentation of this article. The first and third authors of this work are partially supported by MINECO/ FEDER grant number MTM2011–22587. The second author is partially supported by MINECO grant MTM2013-40998-P, an AGAUR Grant Number 2014SGR-568 and the grants FP7-PEOPLE-2012-IRSES 318999 and 316338.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 30203, Cartagena, Región de Murcia, Spain

    Juan L. G. Guirao

  2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193, Barcelona, Catalonia, Spain

    Jaume Llibre

  3. Centro Universitario de la Defensa. Academia General del Aire, Universidad Politécnica de Cartagena, 30720, Santiago de la Ribera, Región de Murcia, Spain

    Juan A. Vera

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  1. Juan L. G. Guirao
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  2. Jaume Llibre
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  3. Juan A. Vera
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Correspondence to Jaume Llibre.

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Guirao, J.L.G., Llibre, J. & Vera, J.A. Periodic orbits of a perturbed 3-dimensional isotropic oscillator with axial symmetry. Nonlinear Dyn 83, 839–848 (2016). https://doi.org/10.1007/s11071-015-2371-z

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  • DOI: https://doi.org/10.1007/s11071-015-2371-z

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