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Determination of nonlinear stability for low order resonances by a geometric criterion

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Abstract

We consider the problem of stability of equilibrium points in Hamiltonian systems of two degrees of freedom under low order resonances. For resonances of order bigger than two there are several results giving stability conditions, in particular one based on the geometry of the phase flow and a set of invariants. In this paper we show that this geometric criterion is still valid for low order resonances, that is, resonances of order two and resonances of order one. This approach provides necessary stability conditions for both the semisimple and non-semisimple cases, with an appropriate choice of invariants.

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References

  1. Alfriend, K. T., Stability and Motion in Two Degree of Freedom Hamiltonian Systems for Two to One Commensurability, Celestial Mech., 1971, vol. 3, no. 2, pp. 247–265.

    Article  MATH  Google Scholar 

  2. Alfriend, K. T., Stability of and Motion about L 4 at Three to One Commensurability, Celestial Mech., 1971, vol. 4, no. 1, pp. 60–77.

    Article  MATH  Google Scholar 

  3. Arnol’d, V. I., The Stability of the Equilibrium Position of a Hamiltonian System of Ordinary Differential Equations in the General Elliptic Case, Dokl. Akad. Nauk SSSR, 1961, vol. 137, pp. 255–257 [Soviet Math. Dokl., 1961, vol. 2, pp. 247–249].

    MathSciNet  Google Scholar 

  4. Birkhoff, G.D., Dynamical Systems, Amer. Math. Soc. Colloq. Publ., vol. 9, Providence, RI: AMS, 1966.

    MATH  Google Scholar 

  5. Cabral, H.E. and Meyer, K. R., Stability of Equilibria and Fixed Points of Conservative Systems, Nonlinearity, 1999, vol. 12, pp. 1351–1362.

    Article  MathSciNet  MATH  Google Scholar 

  6. Dirichlet, G. L., Über die Stabilität des Gleichgewichts, in Werke: Vol. 2, Berlin: Reimer, 1897, pp. 5–8.

    Google Scholar 

  7. Elipe, A., Complete Reduction of Oscillators in Resonance p: q, Phys. Rev. E, 2000, vol. 61, pp. 6477–6484.

    Article  Google Scholar 

  8. Elipe, A., Lanchares, V., López-Moratalla, T., and Riaguas, A., Nonlinear Stability in Resonant Cases: A Geometrical Approach, J. Nonlinear Sci., 2001, vol. 11, no. 3, pp. 211–222.

    Article  MathSciNet  MATH  Google Scholar 

  9. Elipe, A., Lanchares, V., and Pascual, A. I., On the Stability of Equilibria in Two Degrees of Freedom Hamiltonian Systems Under Resonances, J. Nonlinear Sci., 2005, vol. 15, no. 5, pp. 305–319.

    Article  MathSciNet  MATH  Google Scholar 

  10. Lerman, L.M. and Markova, A.P., On Stability at the Hamiltonian Hopf Bifurcation, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 148–162.

    Article  MathSciNet  MATH  Google Scholar 

  11. Markeev, A. P., Stability of a Canonical System with Two Degrees of Freedom in the Presence of Resonance, Prikl. Mat. Mekh., 1968, vol. 32, no. 4, pp. 738–744 [J. Appl. Math. Mech., 1968, vol. 32, no. 4, pp. 766–772].

    MathSciNet  MATH  Google Scholar 

  12. Markeev, A.P., On a Critical Case of Fourth-Order Resonance in a Hamiltonian System with One Degree of Freedom, Prikl. Mat. Mekh., 1997, vol. 61, no. 3, pp. 369–376 [J. Appl. Math. Mech., 1997, vol. 61, no. 3, pp. 355–361].

    MathSciNet  MATH  Google Scholar 

  13. Markeev, A.P., On the Problem of the Stability of the Equilibrium Position of a Hamiltonian System at Resonance 3: 1, Prikl. Mat. Mekh., 2001, vol. 65, no. 4, pp. 653–660 [J. Appl. Math. Mech., 2001, vol. 65, no. 4, pp. 639–645].

    MathSciNet  MATH  Google Scholar 

  14. Meyer, K. R. and Schmidt, D. S., The Stability of the Lagrange Triangular Point and a Theorem of Arnol’d, J. Differential Equations, 1986, vol. 62, no. 2, pp. 222–236.

    Article  MathSciNet  MATH  Google Scholar 

  15. Palacián, J. and Yanguas, P., Reduction of Polynomial Planar Hamiltonians with Quadratic Unperturbed Part, SIAM Rev., 2000, vol. 42, no. 4, pp. 671–691.

    Article  MathSciNet  MATH  Google Scholar 

  16. Pascual, A. I., Sobre la estabilidad de sistemas hamiltonianos de dos grados de libertad bajo resonancias, Doctoral Thesis, Logroño, La Rioja, Spain, 2005.

  17. Siegel, C. L. and Moser, L. K., Lectures on Celestial Mechanics, New York: Springer, 1971.

    Book  MATH  Google Scholar 

  18. Sokol’ski, A.G., On the Stability of an Autonomous Hamiltonian System with Two Degrees of Freedom in the Case of Equal Frequencies, Prikl. Mat. Mekh., 1974, vol. 38, pp. 791–799 [J. Appl. Math. Mech., 1974, vol. 38, pp. 741–749].

    Google Scholar 

  19. Sokol’ski, A.G., On Stability of an Autonomous Hamiltonian System with Two Degrees of Freedom under First-Order Resonance, Prikl. Mat. Mekh., 1977, vol. 41, no. 1, pp. 24–33 [J. Appl. Math. Mech., 1977, vol. 41, no. 1, pp. 20–28].

    Google Scholar 

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Authors and Affiliations

  1. Departamento Matemáticas y Computación, CIME, Universidad de La Rioja Univ. de La Rioja, 26004, Logroño, Spain

    Víctor Lanchares & Ana I. Pascual

  2. Grupo de Mecánica Espacial-IUMA and Centro Universitario de la Defensa de Zaragoza Univ. de Zaragoza, 50009, Zaragoza, Spain

    Antonio Elipe

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  1. Víctor Lanchares
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  2. Ana I. Pascual
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  3. Antonio Elipe
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Correspondence to Víctor Lanchares.

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Lanchares, V., Pascual, A.I. & Elipe, A. Determination of nonlinear stability for low order resonances by a geometric criterion. Regul. Chaot. Dyn. 17, 307–317 (2012). https://doi.org/10.1134/S1560354712030070

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  • DOI: https://doi.org/10.1134/S1560354712030070

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