American Institute of Mathematical Sciences

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December  2010, 3(4): 517-532. doi: 10.3934/dcdss.2010.3.517

A geometric fractional monodromy theorem

Henk Broer 1, , Konstantinos Efstathiou 1, and  Olga Lukina 2,

1. 

Department of Mathematics, University of Groningen, PO Box 407, 9700 AK, Groningen, Netherlands
2. 

Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom

Received  April 2009 Revised  May 2010 Published  August 2010

We prove the existence of fractional monodromy for two degree of freedom integrable Hamiltonian systems with one-parameter families of curled tori under certain general conditions. We describe the action coordinates of such systems near curled tori and we show how to compute fractional monodromy using the notion of the rotation number.
Keywords: integrable Hamiltonian systems., Fractional monodromy.
Mathematics Subject Classification: 37J35, 70H06, 70H3.
Citation: Henk Broer, Konstantinos Efstathiou, Olga Lukina. A geometric fractional monodromy theorem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 517-532. doi: 10.3934/dcdss.2010.3.517
References:
[1]

Y. Colin de Verdière and San Vũ Ngoc, Singular Bohr-Sommerfeld rules for 2D integrable systems,, Ann. Sci. Éc. Norm. Sup., 36 (2003), 1.   Google Scholar

[2]

R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy,, J. Diff. Eqs., 172 (2001), 42.  doi: 10.1006/jdeq.2000.3852.  Google Scholar

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R. H. Cushman, H. Dullin, H. Hanßmann and S. Schmidt, The 1:±2 resonance,, Regular and Chaotic Dynamics, 12 (2007), 642.  doi: 10.1134/S156035470706007X.  Google Scholar

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R. H. Cushman and San Vũ Ngoc, Sign of the monodromy for Liouville integrable systems,, Annales Henri Poincaré, 3 (2002), 883.  doi: 10.1007/s00023-002-8640-7.  Google Scholar

[5]

R. Devaney, "An Introduction to Chaotic Dynamical Systems,'', Benjamin-Cummings, (1986).   Google Scholar

[6]

J. J. Duistermaat, On global action-angle coordinates,, Comm. Pure Appl. Math., 33 (1980), 687.  doi: 10.1002/cpa.3160330602.  Google Scholar

[7]

K. Efstathiou, R. H. Cushman and D. A. Sadovskií, Fractional monodromy in the 1:-2 resonance,, Advances in Mathematics, 209 (2007), 241.  doi: 10.1016/j.aim.2006.05.006.  Google Scholar

[8]

A. Giacobbe, Fractional monodromy: Parallel transport of homology cycles,, Diff. Geom. and Appl., 26 (2008), 140.  doi: 10.1016/j.difgeo.2007.11.011.  Google Scholar

[9]

O. V. Lukina, "Geometry of Torus Bundles in Integrable Hamiltonian Systems,'', Ph.D thesis, (2008).   Google Scholar

[10]

O. V. Lukina, F. Takens and H. W. Broer, Global properties of integrable Hamiltonian systems,, Regular and Chaotic Dynamics, 13 (2008), 602.  doi: 10.1134/S1560354708060105.  Google Scholar

[11]

N. N. Nekhoroshev, Action-angle variables and their generalizations,, Trans. Moscow Math. Soc., 26 (1972), 180.   Google Scholar

[12]

N. N. Nekhoroshev, Fractional monodromy in the case of arbitrary resonances,, Sbornik: Mathematics, 198 (2007), 383.  doi: 10.1070/SM2007v198n03ABEH003841.  Google Scholar

[13]

N. N. Nekhoroshev, Fuzzy fractional monodromy and the section-hyperboloid,, Milan J. Math., 76 (2008), 1.  doi: 10.1007/s00032-008-0085-0.  Google Scholar

[14]

N. N. Nekhoroshev, D. A. Sadovskií and B. I. Zhilinskií, Fractional monodromy of resonant classical and quantum oscillators,, Comptes Rendus Acad. Sci. Paris, 335 (2002), 985.   Google Scholar

[15]

N. N. Nekhoroshev, D. A. Sadovskií and B. I. Zhilinskií, Fractional Hamiltonian monodromy,, Annales Henri Poincaré, 7 (2006), 1099.  doi: 10.1007/s00023-006-0278-4.  Google Scholar

[16]

D. Sugny, P. Mardešić, M. Pelletier, A. Jebrane and H. R. Jauslin, Fractional Hamiltonian monodromy from a Gauss-Manin monodromy,, J. Math. Phys., 49 (2008), 042701.  doi: 10.1063/1.2863614.  Google Scholar

[17]

San Vũ Ngoc, Quantum monodromy in integrable systems,, Communications in Mathematical Physics, 203 (1999), 465.  doi: 10.1007/s002200050621.  Google Scholar

[18]

N. T. Zung, A note on focus-focus singularities,, Diff. Geom. and Appl., 7 (1997), 123.  doi: 10.1016/S0926-2245(96)00042-3.  Google Scholar

[19]

Nguyen Tien Zung, Symplectic topology of integrable Hamiltonian systems, I: Arnold-Liouville with singularities,, Comp. Math., 101 (1996), 179.   Google Scholar

show all references

References:
[1]

Y. Colin de Verdière and San Vũ Ngoc, Singular Bohr-Sommerfeld rules for 2D integrable systems,, Ann. Sci. Éc. Norm. Sup., 36 (2003), 1.   Google Scholar

[2]

R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy,, J. Diff. Eqs., 172 (2001), 42.  doi: 10.1006/jdeq.2000.3852.  Google Scholar

[3]

R. H. Cushman, H. Dullin, H. Hanßmann and S. Schmidt, The 1:±2 resonance,, Regular and Chaotic Dynamics, 12 (2007), 642.  doi: 10.1134/S156035470706007X.  Google Scholar

[4]

R. H. Cushman and San Vũ Ngoc, Sign of the monodromy for Liouville integrable systems,, Annales Henri Poincaré, 3 (2002), 883.  doi: 10.1007/s00023-002-8640-7.  Google Scholar

[5]

R. Devaney, "An Introduction to Chaotic Dynamical Systems,'', Benjamin-Cummings, (1986).   Google Scholar

[6]

J. J. Duistermaat, On global action-angle coordinates,, Comm. Pure Appl. Math., 33 (1980), 687.  doi: 10.1002/cpa.3160330602.  Google Scholar

[7]

K. Efstathiou, R. H. Cushman and D. A. Sadovskií, Fractional monodromy in the 1:-2 resonance,, Advances in Mathematics, 209 (2007), 241.  doi: 10.1016/j.aim.2006.05.006.  Google Scholar

[8]

A. Giacobbe, Fractional monodromy: Parallel transport of homology cycles,, Diff. Geom. and Appl., 26 (2008), 140.  doi: 10.1016/j.difgeo.2007.11.011.  Google Scholar

[9]

O. V. Lukina, "Geometry of Torus Bundles in Integrable Hamiltonian Systems,'', Ph.D thesis, (2008).   Google Scholar

[10]

O. V. Lukina, F. Takens and H. W. Broer, Global properties of integrable Hamiltonian systems,, Regular and Chaotic Dynamics, 13 (2008), 602.  doi: 10.1134/S1560354708060105.  Google Scholar

[11]

N. N. Nekhoroshev, Action-angle variables and their generalizations,, Trans. Moscow Math. Soc., 26 (1972), 180.   Google Scholar

[12]

N. N. Nekhoroshev, Fractional monodromy in the case of arbitrary resonances,, Sbornik: Mathematics, 198 (2007), 383.  doi: 10.1070/SM2007v198n03ABEH003841.  Google Scholar

[13]

N. N. Nekhoroshev, Fuzzy fractional monodromy and the section-hyperboloid,, Milan J. Math., 76 (2008), 1.  doi: 10.1007/s00032-008-0085-0.  Google Scholar

[14]

N. N. Nekhoroshev, D. A. Sadovskií and B. I. Zhilinskií, Fractional monodromy of resonant classical and quantum oscillators,, Comptes Rendus Acad. Sci. Paris, 335 (2002), 985.   Google Scholar

[15]

N. N. Nekhoroshev, D. A. Sadovskií and B. I. Zhilinskií, Fractional Hamiltonian monodromy,, Annales Henri Poincaré, 7 (2006), 1099.  doi: 10.1007/s00023-006-0278-4.  Google Scholar

[16]

D. Sugny, P. Mardešić, M. Pelletier, A. Jebrane and H. R. Jauslin, Fractional Hamiltonian monodromy from a Gauss-Manin monodromy,, J. Math. Phys., 49 (2008), 042701.  doi: 10.1063/1.2863614.  Google Scholar

[17]

San Vũ Ngoc, Quantum monodromy in integrable systems,, Communications in Mathematical Physics, 203 (1999), 465.  doi: 10.1007/s002200050621.  Google Scholar

[18]

N. T. Zung, A note on focus-focus singularities,, Diff. Geom. and Appl., 7 (1997), 123.  doi: 10.1016/S0926-2245(96)00042-3.  Google Scholar

[19]

Nguyen Tien Zung, Symplectic topology of integrable Hamiltonian systems, I: Arnold-Liouville with singularities,, Comp. Math., 101 (1996), 179.   Google Scholar

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