
-
Previous Article
The Euler-Poisson equations: An elementary approach to integrability conditions
- JGM Home
- This Issue
-
Next Article
Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction
A family of compact semitoric systems with two focus-focus singularities
1. | University of Antwerp, Department of Mathematics and Computer Science, Middelheimlaan 1, B-2020 Antwerpen, Belgium |
2. | Rutgers University, Department of Mathematics, Hill Center - Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA |
About 6 years ago, semitoric systems were classified by Pelayo & Vũ Ngọc by means of five invariants. Standard examples are the coupled spin oscillator on $\mathbb{S}^2 \times \mathbb{R}^2$ and coupled angular momenta on $\mathbb{S}^2 \times \mathbb{S}^2$, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a $6$-parameter family of integrable systems on $\mathbb{S}^2 \times \mathbb{S}^2$ and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.
References:
[1] |
J. Alonso, H. Dullin and S. Hohloch, Taylor series and twisting-index invariants of coupled spin-oscillators, arXiv: 1712.06402 (to appear in the Journal of Geometry and Physics). |
[2] |
J. Alonso, H. Dullin and S. Hohloch, Symplectic classification of coupled angular momenta, arXiv: 1808.05849. |
[3] |
M. F. Atiyah,
Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15.
doi: 10.1112/blms/14.1.1. |
[4] |
A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems, Chapman & Hall/CRC, Boca Raton, FL, 2004, Geometry, topology, classification, Translated from the 1999 Russian original.
doi: 10.1201/9780203643426. |
[5] |
P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian bifurcations, in Geometric Mechanics and Symmetry, vol. 306 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2005, 357-402, Based on lectures by Montaldi.
doi: 10.1017/CBO9780511526367.007. |
[6] |
R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, 2nd edition, Birkhäuser/Springer, Basel, 2015, URL https://doi.org/10.1007/978-3-0348-0918-4.
doi: 10.1007/978-3-0348-0918-4. |
[7] |
R. Cushman and V. N. San,
Sign of the monodromy for Liouville integrable systems, Ann. Henri Poincaré, 3 (2002), 883-894.
doi: 10.1007/s00023-002-8640-7. |
[8] |
T. Delzant,
Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France, 116 (1988), 315-339.
doi: 10.24033/bsmf.2100. |
[9] |
H. Dullin and Á. Pelayo,
Generating hyperbolic singularities in semitoric systems via Hopf bifurcations, J. Nonlinear Sci., 26 (2016), 787-811.
doi: 10.1007/s00332-016-9290-0. |
[10] |
K. Efstathiou and N. Martynchuk,
Monodromy of Hamiltonian systems with complexity 1 torus actions, J. Geom. Phys., 115 (2017), 104-115.
doi: 10.1016/j.geomphys.2016.05.014. |
[11] |
L. H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, PhD thesis, University of Stockholm, 1984. |
[12] |
M. Gaudin,
Diagonalisation d'une classe d'hamiltoniens de spin, J. Phys. France, 37 (1976), 1087-1098.
doi: 10.1051/jphys:0197600370100108700. |
[13] |
V. Guillemin and S. Sternberg,
Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513.
doi: 10.1007/BF01398933. |
[14] |
S. Hohloch, S. Sabatini and D. Sepe,
From compact semi-toric systems to Hamiltonian $S^1$-spaces, Discrete Contin. Dyn. Syst., 35 (2015), 247-281.
|
[15] |
S. Hohloch, S. Sabatini, D. Sepe and M. Symington, Faithful semitoric systems, SIGMA, 14 (2018), 084, 66 pages. |
[16] |
D. M. Kane, J. Palmer and Á. Pelayo, Classifying toric and semitoric fans by lifting equations from $\rm{SL}_2(\mathbb{Z})$, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018), Paper No. 016, 43 pp.
doi: 10.3842/SIGMA.2018.016. |
[17] |
D. M. Kane, J. Palmer and Á. Pelayo,
Minimal models of compact symplectic semitoric manifolds, J. Geom. Phys., 125 (2018), 49-74.
doi: 10.1016/j.geomphys.2017.12.005. |
[18] |
Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc., 141 (1999), ⅷ+71pp.
doi: 10.1090/memo/0672. |
[19] |
Y. Le Floch and Á. Pelayo, Symplectic geometry and spectral properties of classical and quantum coupled angular momenta, arXiv: 1607.05419. |
[20] |
Y. Le Floch, Á. Pelayo and S. Vũ Ngọc,
Inverse spectral theory for semiclassical Jaynes-Cummings systems, Math. Ann., 364 (2016), 1393-1413.
doi: 10.1007/s00208-015-1259-z. |
[21] |
J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, New York, 1999, https://doi.org/10.1007/978-0-387-21792-5, A basic exposition of classical mechanical systems.
doi: 10.1007/978-0-387-21792-5. |
[22] |
E. Miranda and N. T. Zung,
Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. École Norm. Sup. (4), 37 (2004), 819-839.
doi: 10.1016/j.ansens.2004.10.001. |
[23] |
J. Palmer,
Moduli spaces of semitoric systems, J. Geom. Phys., 115 (2017), 191-217.
doi: 10.1016/j.geomphys.2017.02.008. |
[24] |
Á. Pelayo,
Hamiltonian and symplectic symmetries: An introduction, Bull. Amer. Math. Soc. (N.S.), 54 (2017), 383-436.
doi: 10.1090/bull/1572. |
[25] |
Á. Pelayo, T. Ratiu and S. Vũ Ngọc,
The affine invariant of proper semitoric integrable systems, Nonlinearity, 30 (2017), 3993-4028.
doi: 10.1088/1361-6544/aa8aec. |
[26] |
Á. Pelayo and S. Vũ Ngọc,
Semitoric integrable systems on symplectic 4-manifolds, Invent. Math., 177 (2009), 571-597.
doi: 10.1007/s00222-009-0190-x. |
[27] |
Á. Pelayo and S. Vũ Ngọc,
Constructing integrable systems of semitoric type, Acta Math., 206 (2011), 93-125.
doi: 10.1007/s11511-011-0060-4. |
[28] |
Á. Pelayo and S. Vũ Ngọc, Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N. S.), 48 (2011), 409-455.
doi: 10.1090/S0273-0979-2011-01338-6. |
[29] |
Á. Pelayo and S. Vũ Ngọc,
Hamiltonian dynamics and spectral theory for spin-oscillators, Comm. Math. Phys., 309 (2012), 123-154.
doi: 10.1007/s00220-011-1360-4. |
[30] |
M. Petrera, Integrable Extensions and Discretizations of Classical Gaudin Models, PhD thesis, Dipartimento di Fisica, Universitá degli Studi di Roma Tre, 2007. |
[31] |
T. Ratiu, C. Wacheux and N. T. Zung, Convexity of singular affine structures and toric-focus integrable hamiltonian systems, arXiv: 1706.01093. |
[32] |
D. A. Sadovskií and B. I. Zĥilinskií,
Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A, 256 (1999), 235-244.
doi: 10.1016/S0375-9601(99)00229-7. |
[33] |
M. Symington, Four dimensions from two in symplectic topology, in Topology and geometry of manifolds (Athens, GA, 2001), vol. 71 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2003,153-208.
doi: 10.1090/pspum/071/2024634. |
[34] |
S. Vũ Ngọc,
On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380.
doi: 10.1016/S0040-9383(01)00026-X. |
[35] |
S. Vũ Ngọc, Systémes Intégrables Semi-Classiques: Du Local au Global, vol. 22 of Panoramas et Synthéses [Panoramas and Syntheses], Société Mathématique de France, Paris, 2006. |
[36] |
S. Vũ Ngọc,
Moment polytopes for symplectic manifolds with monodromy, Adv. Math., 208 (2007), 909-934.
doi: 10.1016/j.aim.2006.04.004. |
[37] |
C. Wacheux, Systémes Intégrables Semi-toriques et Polytopes Moment, PhD thesis, Université de Rennes 1, 2013. |
[38] |
J. Williamson,
On the algebraic problem concerning the normal form of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163.
doi: 10.2307/2371062. |
show all references
References:
[1] |
J. Alonso, H. Dullin and S. Hohloch, Taylor series and twisting-index invariants of coupled spin-oscillators, arXiv: 1712.06402 (to appear in the Journal of Geometry and Physics). |
[2] |
J. Alonso, H. Dullin and S. Hohloch, Symplectic classification of coupled angular momenta, arXiv: 1808.05849. |
[3] |
M. F. Atiyah,
Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15.
doi: 10.1112/blms/14.1.1. |
[4] |
A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems, Chapman & Hall/CRC, Boca Raton, FL, 2004, Geometry, topology, classification, Translated from the 1999 Russian original.
doi: 10.1201/9780203643426. |
[5] |
P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian bifurcations, in Geometric Mechanics and Symmetry, vol. 306 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2005, 357-402, Based on lectures by Montaldi.
doi: 10.1017/CBO9780511526367.007. |
[6] |
R. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, 2nd edition, Birkhäuser/Springer, Basel, 2015, URL https://doi.org/10.1007/978-3-0348-0918-4.
doi: 10.1007/978-3-0348-0918-4. |
[7] |
R. Cushman and V. N. San,
Sign of the monodromy for Liouville integrable systems, Ann. Henri Poincaré, 3 (2002), 883-894.
doi: 10.1007/s00023-002-8640-7. |
[8] |
T. Delzant,
Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France, 116 (1988), 315-339.
doi: 10.24033/bsmf.2100. |
[9] |
H. Dullin and Á. Pelayo,
Generating hyperbolic singularities in semitoric systems via Hopf bifurcations, J. Nonlinear Sci., 26 (2016), 787-811.
doi: 10.1007/s00332-016-9290-0. |
[10] |
K. Efstathiou and N. Martynchuk,
Monodromy of Hamiltonian systems with complexity 1 torus actions, J. Geom. Phys., 115 (2017), 104-115.
doi: 10.1016/j.geomphys.2016.05.014. |
[11] |
L. H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, PhD thesis, University of Stockholm, 1984. |
[12] |
M. Gaudin,
Diagonalisation d'une classe d'hamiltoniens de spin, J. Phys. France, 37 (1976), 1087-1098.
doi: 10.1051/jphys:0197600370100108700. |
[13] |
V. Guillemin and S. Sternberg,
Convexity properties of the moment mapping, Invent. Math., 67 (1982), 491-513.
doi: 10.1007/BF01398933. |
[14] |
S. Hohloch, S. Sabatini and D. Sepe,
From compact semi-toric systems to Hamiltonian $S^1$-spaces, Discrete Contin. Dyn. Syst., 35 (2015), 247-281.
|
[15] |
S. Hohloch, S. Sabatini, D. Sepe and M. Symington, Faithful semitoric systems, SIGMA, 14 (2018), 084, 66 pages. |
[16] |
D. M. Kane, J. Palmer and Á. Pelayo, Classifying toric and semitoric fans by lifting equations from $\rm{SL}_2(\mathbb{Z})$, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018), Paper No. 016, 43 pp.
doi: 10.3842/SIGMA.2018.016. |
[17] |
D. M. Kane, J. Palmer and Á. Pelayo,
Minimal models of compact symplectic semitoric manifolds, J. Geom. Phys., 125 (2018), 49-74.
doi: 10.1016/j.geomphys.2017.12.005. |
[18] |
Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc., 141 (1999), ⅷ+71pp.
doi: 10.1090/memo/0672. |
[19] |
Y. Le Floch and Á. Pelayo, Symplectic geometry and spectral properties of classical and quantum coupled angular momenta, arXiv: 1607.05419. |
[20] |
Y. Le Floch, Á. Pelayo and S. Vũ Ngọc,
Inverse spectral theory for semiclassical Jaynes-Cummings systems, Math. Ann., 364 (2016), 1393-1413.
doi: 10.1007/s00208-015-1259-z. |
[21] |
J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, New York, 1999, https://doi.org/10.1007/978-0-387-21792-5, A basic exposition of classical mechanical systems.
doi: 10.1007/978-0-387-21792-5. |
[22] |
E. Miranda and N. T. Zung,
Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. École Norm. Sup. (4), 37 (2004), 819-839.
doi: 10.1016/j.ansens.2004.10.001. |
[23] |
J. Palmer,
Moduli spaces of semitoric systems, J. Geom. Phys., 115 (2017), 191-217.
doi: 10.1016/j.geomphys.2017.02.008. |
[24] |
Á. Pelayo,
Hamiltonian and symplectic symmetries: An introduction, Bull. Amer. Math. Soc. (N.S.), 54 (2017), 383-436.
doi: 10.1090/bull/1572. |
[25] |
Á. Pelayo, T. Ratiu and S. Vũ Ngọc,
The affine invariant of proper semitoric integrable systems, Nonlinearity, 30 (2017), 3993-4028.
doi: 10.1088/1361-6544/aa8aec. |
[26] |
Á. Pelayo and S. Vũ Ngọc,
Semitoric integrable systems on symplectic 4-manifolds, Invent. Math., 177 (2009), 571-597.
doi: 10.1007/s00222-009-0190-x. |
[27] |
Á. Pelayo and S. Vũ Ngọc,
Constructing integrable systems of semitoric type, Acta Math., 206 (2011), 93-125.
doi: 10.1007/s11511-011-0060-4. |
[28] |
Á. Pelayo and S. Vũ Ngọc, Symplectic theory of completely integrable Hamiltonian systems, Bull. Amer. Math. Soc. (N. S.), 48 (2011), 409-455.
doi: 10.1090/S0273-0979-2011-01338-6. |
[29] |
Á. Pelayo and S. Vũ Ngọc,
Hamiltonian dynamics and spectral theory for spin-oscillators, Comm. Math. Phys., 309 (2012), 123-154.
doi: 10.1007/s00220-011-1360-4. |
[30] |
M. Petrera, Integrable Extensions and Discretizations of Classical Gaudin Models, PhD thesis, Dipartimento di Fisica, Universitá degli Studi di Roma Tre, 2007. |
[31] |
T. Ratiu, C. Wacheux and N. T. Zung, Convexity of singular affine structures and toric-focus integrable hamiltonian systems, arXiv: 1706.01093. |
[32] |
D. A. Sadovskií and B. I. Zĥilinskií,
Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A, 256 (1999), 235-244.
doi: 10.1016/S0375-9601(99)00229-7. |
[33] |
M. Symington, Four dimensions from two in symplectic topology, in Topology and geometry of manifolds (Athens, GA, 2001), vol. 71 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2003,153-208.
doi: 10.1090/pspum/071/2024634. |
[34] |
S. Vũ Ngọc,
On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380.
doi: 10.1016/S0040-9383(01)00026-X. |
[35] |
S. Vũ Ngọc, Systémes Intégrables Semi-Classiques: Du Local au Global, vol. 22 of Panoramas et Synthéses [Panoramas and Syntheses], Société Mathématique de France, Paris, 2006. |
[36] |
S. Vũ Ngọc,
Moment polytopes for symplectic manifolds with monodromy, Adv. Math., 208 (2007), 909-934.
doi: 10.1016/j.aim.2006.04.004. |
[37] |
C. Wacheux, Systémes Intégrables Semi-toriques et Polytopes Moment, PhD thesis, Université de Rennes 1, 2013. |
[38] |
J. Williamson,
On the algebraic problem concerning the normal form of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163.
doi: 10.2307/2371062. |







[1] |
Lingling Liu, Bo Gao, Dongmei Xiao, Weinian Zhang. Identification of focus and center in a 3-dimensional system. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 485-522. doi: 10.3934/dcdsb.2014.19.485 |
[2] |
Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789 |
[3] |
Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228 |
[4] |
Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81 |
[5] |
Qiaoyi Hu, Zhijun Qiao. Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103 |
[6] |
Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171 |
[7] |
Xiuting Li, Lei Zhang. The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3301-3325. doi: 10.3934/dcds.2017140 |
[8] |
Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883 |
[9] |
Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325 |
[10] |
Miguel-C. Muñoz-Lecanda. On some aspects of the geometry of non integrable distributions and applications. Journal of Geometric Mechanics, 2018, 10 (4) : 445-465. doi: 10.3934/jgm.2018017 |
[11] |
Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61 |
[12] |
Carles Simó, Dmitry Treschev. Stability islands in the vicinity of separatrices of near-integrable symplectic maps. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 681-698. doi: 10.3934/dcdsb.2008.10.681 |
[13] |
V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277 |
[14] |
Eric Benoît. Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 911-929. doi: 10.3934/dcdss.2009.2.911 |
[15] |
Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006 |
[16] |
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 |
[17] |
Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227 |
[18] |
Colin Rogers, Tommaso Ruggeri. q-Gaussian integrable Hamiltonian reductions in anisentropic gasdynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2297-2312. doi: 10.3934/dcdsb.2014.19.2297 |
[19] |
Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 587-604. doi: 10.3934/dcds.2008.22.587 |
[20] |
Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67 |
2017 Impact Factor: 0.561
Tools
Metrics
Other articles
by authors
[Back to Top]