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

• Crespo, F ^[1] ; Espejo, D E ^[1] ; Vidal, Claudio ^[1]
  1. [1] Universidad del Bío-Bío

     Universidad del Bío-Bío

     Comuna de Concepción, Chile

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00354-9
• Enlaces
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• Resumen

  • 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.

• Referencias bibliográficas
  • 1. Bardin, B., Lanchares, V.: Conditions of instability of hamiltonian system in case of degeneration. In: Materialen zum wissenschaftlichen...
  • 2. Belitskiy, G.R.: Normal Forms, Invariants and Local Mappings. Kiev (1979)
  • 3. Bruno, A.: Power Geometry in Algebraic and Differential Equations. North-Holland Mathematical Library. Elsevier Science, Amsterdam (2000)
  • 4. Carcamo, D., Vidal, C.: Instability of equilibrium solutions of Hamiltonian systems with n-degrees of freedom under the existence of a...
  • 5. Cleary, P.W.: Nonexistence and existence of various order integrals for two-and three-dimensional polynomial potentials. J. Math. Phys....
  • 6. Deprit, A.: Canonical transformations depending on a small parameter. Celest. Mech. 1(1), 12–30 (1969)
  • 7. Falconi, M., Lacomba, E.A., Vidal, C.: The flow of classical mechanical cubic potential systems. Discrete Contin. Dyn. Syst. A 11(4), 827–842...
  • 8. Goltser Y.M.: Stability and oscillations of parametrically perturbed resonance systems. Doctoral thesis. The Ural Scientific Center, The...
  • 9. Goltser, Y.M.: On the stability of differential equation systems with the spectrum on the imaginary axis. Funct. Differ. Equ. 4(1–2), 47–64...
  • 10. Goltser, Y.M.: Invariant rays and Liapunov functions. Funct. Differ. Equ. 6, 91–110 (1999)
  • 11. Goltser, Y.M.: Stability of mappings with a spectrum on a unit circle. J. Math. Anal. Appl. 196, 841–860 (1995)
  • 12. Goltser, Y.M.: The process of normalization and solution of bifurcation problems of the oscillation and stability theory: A synopsis....
  • 13. Gustavson, F.: On constructing formal integrals of a Hamiltonian system near an equilibrium point. Astron. J. 71, 670–686 (1966)
  • 14. Hori, G.: Teory of general perturbation with unspecifed canonical variables. Publ. Astron. Soc. Jpn. 18, 287–296 (1966)
  • 15. Kamenkov, G.V.: Selected papers (1). Nauka, Moscow (1971)
  • 16. Khazin, L.: On the stability of Hamiltonian systems in the presence of resonances: PMM vol. 35, n–3, 1971, pp. 423–431. J. Appl. Math....
  • 17. Khazin, L.: Interaction of third-order resonances in problems of the stability of Hamiltonian systems. J. Appl. Math. Mech. 48(3), 356–360...
  • 18. Khazin, L., Shnol, E.E.: Stability of Critical Equilibrium States. Cultural Politics. Manchester University Press, Manchester (1991)
  • 19. Markeev, A.: Linear Hamiltonian System and Some Applications to the Problem of Stability of Motion of Satellites Relative to the Center...
  • 20. Molchanov, A.: Stability in the case of a neutral linear approximation. Dokl. Akad. Nauk SSSR 141, 24–27 (1961)
  • 21. Sadovskiy, A.P.: Normal forms of the systems of differential equations with non-zero linear parts. Diff. uravneniniya 8, 5 (1982)
  • 22. Sokol’skii, A.: On stability of self-contained Hamiltonian system with two degrees of freedom in the case of zero frequencies. J. Appl....
  • 23. Takens, F.: Singularities of vector fields. Publ. Math. THES 43, 47 (1974)
  • 24. Turmagambetova, K.V.: Normalization of parametrically perturbed systems with a double zero root and its applications. Ph.D. Thesis. Alma-Ata...
  • 25. Veretennikov, V.G.: Stability and oscillations of nonlinear systems (Russian book). Izdatel’stvo Nauka, Moscow (1984)