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Dynamics of Axially Symmetric Perturbed Hamiltonians in 1:1:1 Resonance

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Abstract

We study the dynamics of a family of perturbed three-degree-of-freedom Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry, and then, periodic solutions and KAM 3-tori of the full system are determined from the relative equilibria. Next, the oscillator symmetry is extended by normalisation up to terms of degree 4 in rectangular coordinates; after truncation of higher orders and reduction to the orbit space, some relative equilibria are established and periodic solutions and KAM 3-tori of the original system are obtained. As a third step, the reduction in the two symmetries leads to a one-degree-of-freedom system that is completely analysed in the twice reduced space. All the relative equilibria together with the stability and parametric bifurcations are determined. Moreover, the invariant 2-tori (related to the critical points of the twice reduced space), some periodic solutions and the KAM 3-tori, all corresponding to the full system, are established. Additionally, the bifurcations of equilibria occurring in the twice reduced space are reconstructed as quasi-periodic bifurcations involving 2-tori and periodic solutions of the full system.

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Acknowledgements

We appreciate the valuable remarks made by the anonymous referees.

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Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Avenida Collao 1202, Casilla 5–C, Concepción, VIII Región, Chile

    D. Carrasco, C. Vidal & J. Vidarte

  2. Departamento de Ingeniería Matemática e Informática and Institute for Advanced Materials (INAMAT), Universidad Pública de Navarra, Campus de Arrosadia, 31006, Pamplona, Spain

    J. F. Palacián & P. Yanguas

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  1. D. Carrasco
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  2. J. F. Palacián
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  3. C. Vidal
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  4. J. Vidarte
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  5. P. Yanguas
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Correspondence to J. F. Palacián.

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Communicated by Tudor Stefan Ratiu.

The authors are partially supported by Projects MTM 2011-28227-C02-01 of the Ministry of Science and Innovation of Spain, MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of Spain, and MTM 2017-88137-C2-1-P of the Ministry of Economy, Industry and Competitiveness of Spain. D. Carrasco is also partially supported by Project DIUBB 165708 3/R, Universidad del Bío-Bío, Chile and by FONDECYT Project 1181061, CONICYT (Chile). This paper is part of Jhon Vidarte’s Ph.D. Thesis in the Program “Doctorado en Matemática Aplicada, Universidad del Bío-Bío”.

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Carrasco, D., Palacián, J.F., Vidal, C. et al. Dynamics of Axially Symmetric Perturbed Hamiltonians in 1:1:1 Resonance. J Nonlinear Sci 28, 1293–1359 (2018). https://doi.org/10.1007/s00332-018-9449-y

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