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Normalization and Existence of Invariant Ray Solutions of a 2-DOF Autonomous Hamiltonian System with Null Frequencies

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Abstract

In this work, we consider a generic 2-DOF autonomous Hamiltonian system, which linear part has all eigenvalues equal to zero. For these type of Hamiltonian, we provide, in an explicit and constructive way, the normal form up to the term of the fourth order. Additionally, we analyze the stability of the normalized Hamiltonian in several cases. Precisely, we obtain instability of the truncated Hamiltonian by explicitly building an invariant ray solution.

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Authors and Affiliations

  1. Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, VIII-Región, Chile

    F. Crespo & Claudio Vidal

  2. Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, VIII-Región, Chile

    D. E. Espejo

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  1. F. Crespo
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  2. D. E. Espejo
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  3. Claudio Vidal
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Correspondence to Claudio Vidal.

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Crespo, F., Espejo, D.E. & Vidal, C. Normalization and Existence of Invariant Ray Solutions of a 2-DOF Autonomous Hamiltonian System with Null Frequencies. Qual. Theory Dyn. Syst. 19, 18 (2020). https://doi.org/10.1007/s12346-020-00354-9

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