-
Previous Article
Some KAM applications to Celestial Mechanics
- DCDS-S Home
- This Issue
-
Next Article
Preface
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 |
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. |
[1] |
Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61 |
[2] |
Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020 |
[3] |
Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479 |
[4] |
Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 587-604. doi: 10.3934/dcds.2008.22.587 |
[5] |
Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67 |
[6] |
Marcel Guardia. Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2829-2859. doi: 10.3934/dcds.2013.33.2829 |
[7] |
P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1 |
[8] |
Aristophanes Dimakis, Folkert Müller-Hoissen. Bidifferential graded algebras and integrable systems. Conference Publications, 2009, 2009 (Special) : 208-219. doi: 10.3934/proc.2009.2009.208 |
[9] |
Leo T. Butler. A note on integrable mechanical systems on surfaces. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1873-1878. doi: 10.3934/dcds.2014.34.1873 |
[10] |
Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227 |
[11] |
Colin Rogers, Tommaso Ruggeri. q-Gaussian integrable Hamiltonian reductions in anisentropic gasdynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2297-2312. doi: 10.3934/dcdsb.2014.19.2297 |
[12] |
Sonomi Kakizaki, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura. Conserved quantities of the integrable discrete hungry systems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 889-899. doi: 10.3934/dcdss.2015.8.889 |
[13] |
Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109 |
[14] |
Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001 |
[15] |
Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325 |
[16] |
Boris S. Kruglikov and Vladimir S. Matveev. Vanishing of the entropy pseudonorm for certain integrable systems. Electronic Research Announcements, 2006, 12: 19-28. |
[17] |
Helen Christodoulidi, Andrew N. W. Hone, Theodoros E. Kouloukas. A new class of integrable Lotka–Volterra systems. Journal of Computational Dynamics, 2019, 6 (2) : 223-237. doi: 10.3934/jcd.2019011 |
[18] |
Dong Chen. Positive metric entropy in nondegenerate nearly integrable systems. Journal of Modern Dynamics, 2017, 11: 43-56. doi: 10.3934/jmd.2017003 |
[19] |
Jacques Demongeot, Dan Istrate, Hajer Khlaifi, Lucile Mégret, Carla Taramasco, René Thomas. From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2121-2134. doi: 10.3934/dcdss.2020181 |
[20] |
Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]