Skip to main content
SpringerLink
Log in

Construction of analytic KAM surfaces and effective stability bounds

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

A class of analytic (possibly) time-dependent Hamiltonian systems withd degrees of freedom and the “corresponding” class of area-preserving, twist diffeomorphisms of the plane are considered. Implementing a recent scheme due to Moser, Salamon and Zehnder, we provide a method that allows us to construct “explicitly” KAM surfaces and, hence, to give lower bounds on their breakdown thresholds. We, then, apply this method to the HamiltonianH≡y 2/2+ε(cosx+cos(x−t)) and to the map (y,x)→(y+ε sinx,x+y+ε sinx) obtaining, with the aid of computer-assisted estimations, explicit approximations (within an error of ∼10−5) of the golden-mean KAM surfaces for complex values of ε with |ε| less or equal than, respectively, 0.015 and 0.65. (The experimental numerical values at which such surfaces are expected to disappear are about, respectively, 0.027 and 0.97.) A possible connection between break-down thresholds and singularities in the complex ε-plane is pointed out.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, V.I.: Proof of a Theorem by A.N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv.18, 9 (1963)

    Google Scholar 

  2. Arnold, V.I., Avez, A.: Ergodic problems of classical mechanics. New York: Benjamin 1968

    Google Scholar 

  3. Aubry, S., Le Daeron, P.Y.: The discrete Frenkel-Kontorova model and its extensions. I. Physica8, 381 (1983)

    Google Scholar 

  4. Benettin, G., Galgani, L., Giorgilli, A.: Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory. Part I. Commun. Math. Phys.113, 87 (1987)

    Google Scholar 

  5. Benettin, G., Gallavotti, G.: Stability of motions near resonances in quasi-integrable Hamiltonian systems. J. Stat. Phys.44, 293 (1986)

    Google Scholar 

  6. Braess, D., Zehnder, E.: On the numerical treatment of a small divisor problem. Numer. Math.39, 269 (1982)

    Google Scholar 

  7. Celletti, A., Chierchia, L.: Rigorous estimates for a computer-assisted KAM theory. J. Math. Phys.28, 2078 (1987)

    Google Scholar 

  8. Celletti, A., Falcolini, C., Porzio, A.: Rigorous numerical stability estimates for the existence of KAM tori in a forced pendulum. Ann. Inst. Henri Poincaré47, 85 (1987)

    Google Scholar 

  9. Chirikov, B.V.: A universal instability of many dimensional oscillator systems. Phys. Rep.52, 263 (1979)

    Google Scholar 

  10. Eckmann, J.-P., Wittwer, P.: Computer methods and Borel summability applied to Feigenbaum's equation. Lecture Notes in Physics, Vol. 227. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  11. Eliasson, H.: Absolutely convergent series expansions for quasiperiodic motions, preprint. Univ. of Stockholm (1987)

  12. Escande, D.F.: Stochasticity in classical Hamiltonian systems: Universal aspects. Phys. Rep.121, 165 (1985)

    Google Scholar 

  13. Escande, D.F.: Private communication

  14. Escande, D.F., Doveil, F.: Renormalization method for computing the threshold of the largescale stochastic instability in two degrees of freedom Hamiltonian systems. J. Stat. Phys.26, 257 (1981)

    Google Scholar 

  15. Falcolini, C.: Private communication

  16. Gallavotti, G.: Perturbation theory for classical Hamiltonian systems. In: Scaling and self-similarity in physics. Fröhlich, J. (ed.) PPh. 7, Boston: Birkhäuser 1984

    Google Scholar 

  17. Greene, J.M.: A method for determining a stochastic transition. J. Math. Phys.20, 1183 (1979)

    Google Scholar 

  18. Hénon, M., Heiles, C.: The applicability of the third integral of motion: Some numerical experiments. Astron. J.69, 73 (1964)

    Google Scholar 

  19. Herman, M.: Sur le courbes invariantes par le difféomorphismes de l'anneau. Astérisque2, 144 (1986)

    Google Scholar 

  20. Iooss, G., Helleman, R.H.G., Stora, R. (eds.): Chaotic behaviour of deterministic systems. Amsterdam: North-Holland 1983

    Google Scholar 

  21. Kolmogorov, A.N.: On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Dokl. Akad. Nauk. SSR98, 469 (1954)

    Google Scholar 

  22. Lanford III, O.E.: Computer assisted proofs in analysis. Phys. A124, 465 (1984)

    Google Scholar 

  23. MacKay, R.S.: Transition to chaos for area-preserving maps. Lecture Notes in Physics, Vol. 247, p. 390. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  24. MacKay, R.S., Percival, I.C.: Converse KAM: Theory and practice. Commun. Math. Phys.98, 469 (1985)

    Google Scholar 

  25. Mather, J.N.: Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology21, 457 (1982)

    Google Scholar 

  26. Mather, J.N.: Non-existence of invariant circles. Ergodic Theory Dyn. Syst.4, 301 (1984)

    Google Scholar 

  27. Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nach. Akad. Wiss. Göttingen, Math. Phys. K1. II1, 1 (1962)

    Google Scholar 

  28. Moser, J.: A rapidly convergent iteration method and non-linear partial differential equations. Ann. Scuola Norm. Sup. Pisa20, 265 (1966)

    Google Scholar 

  29. Moser, J.: Minimal solutions of variational problems on a torus. Ann. Inst. Henri Poincaré3, 229 (1986)

    Google Scholar 

  30. Moser, J.: Minimal foliations on a torus. Forschungsinstitut für Mathematik, ETH Zürich, September 1987

  31. Nekhoroshev, N.N.: An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Russ. Math. Surv.32, 1 (1977)

    Google Scholar 

  32. Percival, I.C.: Variational principles for invariant tori and cantori, in nonlinear dynamics and the beam-beam interaction. AIP Conference Proceedings 57, p. 302. M. Month, J.C. Herrera (eds.). (1980)

  33. Rüssmann, H.: On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus. Lecture Notes in Physics, Vol. 38, p. 598. Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

  34. Rüssmann, H.: On the existence of invariant curves of twist mappings of an annulus. Lecture Notes in Mathematics, Vol. 1007, p. 677. Berlin, Heidelberg, New York: Springer 1983

    Google Scholar 

  35. Salamon, D., Zehnder, E.: KAM theory in configuration space. Preprint 1987

  36. Wayne, C.E.: The KAM theory of systems with short range interactions. I. Commun. Math. Phys.96, 311 (1984)

    Google Scholar 

  37. Wayne, C.E.: Bounds on the trajectories of a system of weakly coupled rotators. Commun. Math. Phys.104, 21 (1986)

    Google Scholar 

  38. Wisdom, J., Peale, S.J.: The chaotic rotation of Hyperion. Icarus58, 137 (1984)

    Google Scholar 

  39. (no author listed) Vax architecture handbook. Digital Equipment Corporation (1981)

Download references

Author information

Author notes

  1. Luigi Chierchia

    Present address: Dipartimento di Matematica, II Università di Roma, I-00173, Roma, Italy

Authors and Affiliations

  1. Forschungsinstitut für Mathematik, ETH-Zentrum, CH-8092, Zürich, Switzerland

    Alessandra Celletti & Luigi Chierchia

Authors
  1. Alessandra Celletti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luigi Chierchia
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by J.-P. Eckmann

To our friend and colleague Paola Calderoni

Supported by Consiglio Nazionale delle Ricerche, Italy

Rights and permissions

Reprints and permissions

About this article

Cite this article

Celletti, A., Chierchia, L. Construction of analytic KAM surfaces and effective stability bounds. Commun.Math. Phys. 118, 119–161 (1988). https://doi.org/10.1007/BF01218480

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01218480

Keywords

Search

Navigation