### Abstract

© 2015, Springer Science+Business Media Dordrecht. We study the periodic orbits of a generalized Yang–Mills Hamiltonian H depending on a parameter β. Playing with the parameter β 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 H = h > 0 the Hamiltonian system has at least four periodic solutions if either β < 0, or (Formula Presented.); and at least 12 periodic solutions if (Formula Presented.) and (Formula Presented.). We also study the linear stability or instability of these periodic solutions.

Original language | English |
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Pages (from-to) | 839-848 |

Journal | Nonlinear Dynamics |

Volume | 83 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 1 Jan 2016 |

### Keywords

- 3D isotropic oscillators
- 3D Yang–Mills Hamiltonian
- Averaging theory
- Periodic orbits
- Stability of periodic orbits

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## Cite this

Guirao, J. L. G., Llibre, J., & Vera, J. A. (2016). Periodic orbits of a perturbed 3-dimensional isotropic oscillator with axial symmetry.

*Nonlinear Dynamics*,*83*(1-2), 839-848. https://doi.org/10.1007/s11071-015-2371-z