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

A geometric fractional monodromy theorem

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.
Citation: Henk Broer, Konstantinos Efstathiou, Olga Lukina. A geometric fractional monodromy theorem. Discrete and 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), 1-55.

[2]

R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy, J. Diff. Eqs., 172 (2001), 42-58. doi: 10.1006/jdeq.2000.3852.

[3]

R. H. Cushman, H. Dullin, H. Hanßmann and S. Schmidt, The 1:±2 resonance, Regular and Chaotic Dynamics, 12 (2007), 642-663. doi: 10.1134/S156035470706007X.

[4]

R. H. Cushman and San Vũ Ngoc, Sign of the monodromy for Liouville integrable systems, Annales Henri Poincaré, 3 2002, 883-894. doi: 10.1007/s00023-002-8640-7.

[5]

R. Devaney, "An Introduction to Chaotic Dynamical Systems,'' Benjamin-Cummings, Menlo Park, CA, 1986.

[6]

J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.

[7]

K. Efstathiou, R. H. Cushman and D. A. Sadovskií, Fractional monodromy in the 1:-2 resonance, Advances in Mathematics, 209 (2007), 241-273. doi: 10.1016/j.aim.2006.05.006.

[8]

A. Giacobbe, Fractional monodromy: Parallel transport of homology cycles, Diff. Geom. and Appl., 26 (2008), 140-150. doi: 10.1016/j.difgeo.2007.11.011.

[9]

O. V. Lukina, "Geometry of Torus Bundles in Integrable Hamiltonian Systems,'' Ph.D thesis, University of Groningen, 2008.

[10]

O. V. Lukina, F. Takens and H. W. Broer, Global properties of integrable Hamiltonian systems, Regular and Chaotic Dynamics, 13 (2008), 602-644. doi: 10.1134/S1560354708060105.

[11]

N. N. Nekhoroshev, Action-angle variables and their generalizations, Trans. Moscow Math. Soc., 26 (1972), 180-198.

[12]

N. N. Nekhoroshev, Fractional monodromy in the case of arbitrary resonances, Sbornik: Mathematics, 198 (2007), 383-424. doi: 10.1070/SM2007v198n03ABEH003841.

[13]

N. N. Nekhoroshev, Fuzzy fractional monodromy and the section-hyperboloid, Milan J. Math., 76 (2008), 1-14. doi: 10.1007/s00032-008-0085-0.

[14]

N. N. Nekhoroshev, D. A. Sadovskií and B. I. Zhilinskií, Fractional monodromy of resonant classical and quantum oscillators, Comptes Rendus Acad. Sci. Paris, Sér. I, 335 (2002), 985-988.

[15]

N. N. Nekhoroshev, D. A. Sadovskií and B. I. Zhilinskií, Fractional Hamiltonian monodromy, Annales Henri Poincaré, 7 (2006), 1099-1211. doi: 10.1007/s00023-006-0278-4.

[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), 042701-35. doi: 10.1063/1.2863614.

[17]

San Vũ Ngoc, Quantum monodromy in integrable systems, Communications in Mathematical Physics, 203 (1999), 465-479. doi: 10.1007/s002200050621.

[18]

N. T. Zung, A note on focus-focus singularities, Diff. Geom. and Appl., 7 (1997), 123-130. doi: 10.1016/S0926-2245(96)00042-3.

[19]

Nguyen Tien Zung, Symplectic topology of integrable Hamiltonian systems, I: Arnold-Liouville with singularities, Comp. Math., 101 (1996), 179-215.

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), 1-55.

[2]

R. H. Cushman and J. J. Duistermaat, Non-Hamiltonian monodromy, J. Diff. Eqs., 172 (2001), 42-58. doi: 10.1006/jdeq.2000.3852.

[3]

R. H. Cushman, H. Dullin, H. Hanßmann and S. Schmidt, The 1:±2 resonance, Regular and Chaotic Dynamics, 12 (2007), 642-663. doi: 10.1134/S156035470706007X.

[4]

R. H. Cushman and San Vũ Ngoc, Sign of the monodromy for Liouville integrable systems, Annales Henri Poincaré, 3 2002, 883-894. doi: 10.1007/s00023-002-8640-7.

[5]

R. Devaney, "An Introduction to Chaotic Dynamical Systems,'' Benjamin-Cummings, Menlo Park, CA, 1986.

[6]

J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.

[7]

K. Efstathiou, R. H. Cushman and D. A. Sadovskií, Fractional monodromy in the 1:-2 resonance, Advances in Mathematics, 209 (2007), 241-273. doi: 10.1016/j.aim.2006.05.006.

[8]

A. Giacobbe, Fractional monodromy: Parallel transport of homology cycles, Diff. Geom. and Appl., 26 (2008), 140-150. doi: 10.1016/j.difgeo.2007.11.011.

[9]

O. V. Lukina, "Geometry of Torus Bundles in Integrable Hamiltonian Systems,'' Ph.D thesis, University of Groningen, 2008.

[10]

O. V. Lukina, F. Takens and H. W. Broer, Global properties of integrable Hamiltonian systems, Regular and Chaotic Dynamics, 13 (2008), 602-644. doi: 10.1134/S1560354708060105.

[11]

N. N. Nekhoroshev, Action-angle variables and their generalizations, Trans. Moscow Math. Soc., 26 (1972), 180-198.

[12]

N. N. Nekhoroshev, Fractional monodromy in the case of arbitrary resonances, Sbornik: Mathematics, 198 (2007), 383-424. doi: 10.1070/SM2007v198n03ABEH003841.

[13]

N. N. Nekhoroshev, Fuzzy fractional monodromy and the section-hyperboloid, Milan J. Math., 76 (2008), 1-14. doi: 10.1007/s00032-008-0085-0.

[14]

N. N. Nekhoroshev, D. A. Sadovskií and B. I. Zhilinskií, Fractional monodromy of resonant classical and quantum oscillators, Comptes Rendus Acad. Sci. Paris, Sér. I, 335 (2002), 985-988.

[15]

N. N. Nekhoroshev, D. A. Sadovskií and B. I. Zhilinskií, Fractional Hamiltonian monodromy, Annales Henri Poincaré, 7 (2006), 1099-1211. doi: 10.1007/s00023-006-0278-4.

[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), 042701-35. doi: 10.1063/1.2863614.

[17]

San Vũ Ngoc, Quantum monodromy in integrable systems, Communications in Mathematical Physics, 203 (1999), 465-479. doi: 10.1007/s002200050621.

[18]

N. T. Zung, A note on focus-focus singularities, Diff. Geom. and Appl., 7 (1997), 123-130. doi: 10.1016/S0926-2245(96)00042-3.

[19]

Nguyen Tien Zung, Symplectic topology of integrable Hamiltonian systems, I: Arnold-Liouville with singularities, Comp. Math., 101 (1996), 179-215.

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