Communications in Mathematical Physics

, Volume 324, Issue 2, pp 549–588 | Cite as

Uncovering Fractional Monodromy

  • K. EfstathiouEmail author
  • H. W. Broer


The uncovering of the role of monodromy in integrable Hamiltonian fibrations has been one of the major advances in the study of integrable Hamiltonian systems in the past few decades: on one hand monodromy turned out to be the most fundamental obstruction to the existence of global action-angle coordinates while, on the other hand, it provided the correct classical analogue for the interpretation of the structure of quantum joint spectra. Fractional monodromy is a generalization of the concept of monodromy: instead of restricting our attention to the toric part of the fibration we extend our scope to also consider singular fibres. In this paper we analyze fractional monodromy for n 1:(−n 2) resonant Hamiltonian systems with n 1, n 2 coprime natural numbers. We consider, in particular, systems that for n 1, n 2 > 1 contain one-parameter families of singular fibres which are ‘curled tori’. We simplify the geometry of the fibration by passing to an appropriate branched covering. In the branched covering the curled tori and their neighborhood become untwisted thus simplifying the geometry of the fibration: we essentially obtain the same type of generalized monodromy independently of n 1, n 2. Fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system.


Homology Group Parallel Transport Original Space Covering Space Closed Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Arnol’d, V.I.: Mathematical methods of classical mechanics. Volume 60 of Graduate Texts in Mathematics. New York: Springer-Verlag, 2nd edition, 1989, translated by K. Vogtmann and A. WeinsteinGoogle Scholar
  2. 2.
    Arnol’d, V.I., Avez, A.: Ergodic Problems of Classical Mechanics. New York: W.A. Benjamin, Inc., 1968Google Scholar
  3. 3.
    Bolsinov, A.V., Fomenko, A.T.: Integrable Hamiltonian systems : geometry, topology, classification. Boca Raton, FL: Chapman & Hall/CRC, 2004Google Scholar
  4. 4.
    Braaksma B.L.J., Broer H.W., Huitema G.B.: Toward a quasi-periodic bifurcation theory. Mem. AMS 83(421), 83–175 (1990)MathSciNetGoogle Scholar
  5. 5.
    Broer H.W., Efstathiou K., Lukina O.V.: A geometric fractional monodromy theorem. Discrete and Continuous Dynamical Systems - Series S (DCDS-S) 3(4), 517–532 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Broer H.W., Hanßmann H., Jorba À., Villanueva J., Wagener F.: Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach. Nonlinearity 16(5), 1751–1791 (2003)MathSciNetADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Broer H.W., Huitema G.B., Takens F.: Unfoldings of quasi-periodic tori. Mem. AMS 83(421), 1–82 (1990)MathSciNetGoogle Scholar
  8. 8.
    Broer H.W., Vegter G.: Bifurcational aspects of parametric resonance. Dynamics Reported, New Series 1, 1–51 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Broer H.W., Vegter G.: Generic Hopf–Neĭmark–Sacker bifurcations in feed-forward systems. Nonlinearity 21(7), 1547–1578 (2008)MathSciNetADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Colin de Verdière Y., Vũ Ngọc S.: Singular Bohr-Sommerfeld rules for 2D integrable systems. Ann. Sci. Éc. Norm. Sup. 36, 1–55 (2003)zbMATHGoogle Scholar
  11. 11.
    Cushman R.H., Bates L.: Global aspects of classical integrable systems. Basel-Boston, Birkhäuser (1997)CrossRefzbMATHGoogle Scholar
  12. 12.
    Cushman R.H., Dullin H., Hanßmann H., Schmidt S.: The 1:±2 resonance. Regular and Chaotic Dynamics 12(6), 642–663 (2007)MathSciNetADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Cushman, R.H., Knörrer, H.: The energy momentum mapping of the Lagrange top. In: Differential Geometric Methods in Mathematical Physics, Volume 1139 of Lecture Notes in Mathematics, Berlin-Heidelberg-New York: Springer, 1985, pp. 12–24Google Scholar
  14. 14.
    Cushman R.H., Sadovskií D.A.: Monodromy in the hydrogen atom in crossed fields. Physica D 142, 166–196 (2000)MathSciNetADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Davison C.M., Dullin H.R., Bolsinov A.V.: Geodesics on the ellipsoid and monodromy. J. Geom. Phys. 57, 2437–2454 (2007)MathSciNetADSCrossRefzbMATHGoogle Scholar
  16. 16.
    Duistermaat J.J.: On global action-angle coordinates. Comm. Pure Appl. Math. 33, 687–706 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dullin H., Giacobbe A., Cushman R.H.: Monodromy in the resonant swing spring. Physica D 190, 15–37 (2004)MathSciNetADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Efstathiou, K.: Metamorphoses of Hamiltonian systems with symmetries. Volume 1864 of Lecture Notes in Mathematics. Berlin-Heidelberg-New York: Springer-Verlag, 2005Google Scholar
  19. 19.
    Efstathiou K., Cushman R.H., Sadovskií D.A.: Fractional monodromy in the 1:−2 resonance. Adv. Math. 209, 241–273 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Efstathiou K., Giacobbe A.: The topology associated to cusp singular points. Nonlinearity 25(12), 3409–3422 (2012)MathSciNetADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Efstathiou K., Sugny D.: Integrable Hamiltonian systems with swallowtails. J. Phys. A: Math. Theor. 43, 085216 (2010)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Giacobbe A.: Fractional monodromy: parallel transport of homology cycles. Diff. Geom. and Appl. 26, 140–150 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Giacobbe A., Cushman R.H., Sadovskií D.A., Zhilinskií B.I.: Monodromy of the quantum 1:1:2 resonant swing spring. J. Math. Phys. 45, 5076–5100 (2004)MathSciNetADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Hatcher, A.: Notes on basic 3-manifold topology. Available online at, 2000
  25. 25.
    Hatcher A.: Algebraic topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  26. 26.
    Lukina O.V., Takens F., Broer H.W.: Global properties of integrable Hamiltonian systems. Reg. Chaotic Dyn. 13, 602–644 (2008)MathSciNetADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Nekhoroshev N.N.: Fractional monodromy in the case of arbitrary resonances. Sbornik : Math. 198, 383–424 (2007)MathSciNetADSCrossRefzbMATHGoogle Scholar
  28. 28.
    Nekhoroshev N.N.: Fuzzy fractional monodromy and the section-hyperboloid. Milan J. Math. 76, 1–14 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nekhoroshev N.N., Sadovskií D.A., Zhilinskií B.I.: Fractional monodromy of resonant classical and quantum oscillators. Comptes Rendus Math. 335(11), 985–988 (2002)CrossRefzbMATHGoogle Scholar
  30. 30.
    Nekhoroshev N.N., Sadovskií D.A., Zhilinskií B.I.: Fractional Hamiltonian monodromy. Ann. H. Poincaré 7, 1099–1211 (2006)CrossRefzbMATHGoogle Scholar
  31. 31.
    Schmidt S., Dullin H.R.: Dynamics near the p : q resonance. Physica D 239(19), 1884–1891 (2010)MathSciNetADSCrossRefzbMATHGoogle Scholar
  32. 32.
    Sugny, D., Mardešić, P., Pelletier, M., Jebrane, A., Jauslin, H.R.: Fractional Hamiltonian monodromy from a Gauss-Manin monodromy. J. Math. Phys. 49, 042701–35 (2008)Google Scholar
  33. 33.
    Vũ Ngọc S.: Quantum monodromy in integrable systems. Commun. Math. Phys. 203(2), 465–479 (1999)ADSCrossRefGoogle Scholar
  34. 34.
    Waalkens H., Dullin H.R.: Quantum monodromy in prolate ellipsoidal billiards. Ann. Phys. 295, 81–112 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  35. 35.
    Waalkens H., Junge A., Dullin H.R.: Quantum monodromy in the two-centre problem. J. Phys. A 36, L307–L314 (2003)MathSciNetADSCrossRefzbMATHGoogle Scholar
  36. 36.
    Zung N.T.: A note on focus-focus singularities. Diff. Geom. Appl. 7, 123–130 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Tien Zung N.: Symplectic topology of integrable Hamiltonian systems, I: Arnold-Liouville with singularities. Comp. Math. 101, 179–215 (1996)zbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands
  2. 2.Department of Mathematical SciencesXi’an Jiaotong–Liverpool UniversitySuzhouChina

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