Normal form construction for nearly-integrable systems with dissipation

Abstract

We consider a dissipative vector field which is represented by a nearly-integrable Hamiltonian flow to which a dissipative contribution is added. The vector field depends upon two parameters, namely the perturbing and dissipative parameters, and by a drift term. We study an -dimensional, time-dependent vector field, which is motivated by mathematical models in Celestial Mechanics. Assuming to start with non-resonant initial conditions, we provide the construction of the normal form up to an arbitrary order. To construct the normal form, a suitable choice of the drift parameter must be performed. The normal form allows also to provide an explicit expression of the frequency associated to the normalized coordinates. We also give an example in which we construct explicitly the normal form, we make a comparison with a numerical integration, and we determine the parameter values and the time interval of validity of the normal form.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Arnol’d, V. I., Proof of a Theorem of A.N.Kolmogorov on the Invariance of Quasi-periodic Motions under Small Perturbations of the Hamiltonian, Uspekhi Mat. Nauk, 1963, vol. 18, no. 5, pp. 13–40 [Russian Math. Surveys, 1963, vol. 18, no. 5, pp. 9–36].

    Google Scholar 

  2. 2.

    Beaugé, C. and Ferraz-Mello, S., Resonance Trapping in the Primordial Solar Nebula: The Case of a Stokes Drag Dissipation, Icarus, 1993, vol. 103, pp. 301–318.

    Article  Google Scholar 

  3. 3.

    Broer, H.W., Huitema, G.B., and Sevryuk, M.B., Quasi-Periodic Motions in Families of Dynamical Systems: Order Amidst Chaos, Lecture Notes in Math., vol. 1645, Berlin: Springer, 1996.

    Google Scholar 

  4. 4.

    Calleja, R., Celletti, A., and de la Llave, R., KAM-Theory for Conformally Symplectic Systems, Preprint, 2011 (available on the Mathematical Physics Preprint Archive: mp arc 11-188).

  5. 5.

    Celletti, A., Stability and Chaos in Celestial Mechanics, Berlin: Springer, 2010.

    Google Scholar 

  6. 6.

    Celletti, A. and Chierchia, L., Quasi-Periodic Attractors in Celestial Mechanics, Arch. Rat. Mech. Anal., 2009, vol. 191, no. 2, pp. 311–345.

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Celletti, A., Stefanelli, L., Lega, E., and Froeschlé, C., Global Dynamics of the Regularized Restricted Three-Body Problem with Dissipation, Celestial Mech. Dynam. Astronom., 2011, vol. 109, no. 3, pp. 265–284.

    MathSciNet  Article  Google Scholar 

  8. 8.

    Ciocci, M.-C., Litvak-Hinenzon, A., and Broer, H.W., Survey on Dissipative KAM Theory including Quasi-Periodic Bifurcation Theory, Geometric Mechanics and Symmetry: The Peyresq Lectures, J. Montaldi and T. Ratiu (Eds.), London Math. Soc. Lecture Note Ser., vol. 306, Cambridge: Cambridge Univ. Press, 2005, pp. 303–355.

    Google Scholar 

  9. 9.

    Delshams, A., Guillamon, A., and Lázaro, J.T., A Pseudo-Normal Form for Planar Vector Fields, Qual. Theory Dyn. Syst., 2002, vol. 3, no. 1, pp. 51–82.

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Henrard, J., A Survey of Poisson Series Processors, Celestial Mech. Dynam. Astronom., 1989, vol. 45, pp. 245–253.

    Google Scholar 

  11. 11.

    Fassò, F., Lie Series Method for Vector Fields and Hamiltonian Perturbation Theory, Z. Angew. Math. Phys., 1990, vol. 41, pp. 843–864.

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Feudel, U., Grebogi, C., Hunt, B.R., and Yorke, J.A., Map with More Than 100 Coexisting Low-Period Periodic Attractors, Phys. Rev. E, 1996, vol. 54, no. 1, pp. 71–81.

    MathSciNet  Article  Google Scholar 

  13. 13.

    Gallavotti, G., The Elements of Mechanics, New York: Springer, 1983.

    Google Scholar 

  14. 14.

    Iooss, G. and Lombardi, E., Polynomial Normal Forms with Exponentially Small Remainder for Analytic Vector Fields, J. Differential Equations, 2005, vol. 212, no. 1, pp. 1–61.

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Kolmogorov, A.N., On Conservation of Conditionally PeriodicMotions for a Small Change in Hamilton’s Function, Dokl. Akad. Nauk SSSR (N. S.), 1954, vol. 98, pp. 527–530 (Russian) (see also: Stochastic Behaviour in Classical and Quantum Hamiltonian Systems (Volta Memorial Conference, Como, 1977), Lect. Notes Phys. Monogr., vol. 93, Berlin: Springer, 1979, pp. 51–56).

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Markeev, A.P., On the Stability of the Planar Motions of a Rigid Body in the Kovalevskaya Case, Prikl. Mat. Mekh., 2001, vol. 65, no. 1, pp. 51–58 [J. Appl. Math. Mech., 2001, vol. 65, no. 1, pp. 47–54].

    MathSciNet  Google Scholar 

  17. 17.

    Moser, J., On Invariant Curves of Area-Preserving Mappings of an Annulus, Nach. Akad. Wiss. Göttingen, Math. Phys. Kl. II, 1962, vol. 1962, no. 1, pp. 1–20.

    Google Scholar 

  18. 18.

    Moser, J., Convergent Series Expansions for Quasi-Periodic Motions, Math. Ann., 1967, vol. 169, pp. 136–176.

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Nekhoroshev, N. N., An Exponential Estimate of the Stability Time of Near-Integrable Hamiltonian Systems, Uspekhi Mat. Nauk, 1977, vol. 32, no. 6(198), pp. 5–66 [Russian Math. Surveys, 1977, vol. 32, no. 6, pp. 1–65].

    MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Aessandra Celletti.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Celletti, A., Lhotka, C. Normal form construction for nearly-integrable systems with dissipation. Regul. Chaot. Dyn. 17, 273–292 (2012). https://doi.org/10.1134/S1560354712030057

Download citation

MSC2010 numbers

  • 37J35

Keywords

  • dissipative system
  • normal form
  • non-resonant motion