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Traveling Waves Impulses of FitzHugh Model with Diffusion and Cross-Diffusion Read chapter Read first chapter

2016 | OriginalPaper | Chapter

15. Abelian Integrals: From the Tangential 16th Hilbert Problem to the Spherical Pendulum

Authors : Pavao Mardešić, Dominique Sugny, Léo Van Damme

Published in: Mathematical Sciences with Multidisciplinary Applications

Publisher: Springer International Publishing

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Abstract

In this chapter we deal with abelian integrals. They play a key role in the infinitesimal version of the 16th Hilbert problem. Recall that 16th Hilbert problem and its ramifications is one of the principal research subject of Christiane Rousseau and of the first author. We recall briefly the definition and explain the role of abelian integrals in 16th Hilbert problem. We also give a simple well-known proof of a property of abelian integrals. The reason for presenting it here is that it serves as a model for more complicated and more original treatment of abelian integrals in the study of Hamiltonian monodromy of fully integrable systems, which is the main subject of this chapter. We treat in particular the simplest example presenting non-trivial Hamiltonian monodromy: the spherical pendulum.

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Literature
1.
Alber, M.S., Marsden, J.E.: Semiclassical monodromy and the spherical pendulum as a complex Hamiltonian system. In: Conservative Systems and Quantum Chaos. Fields Institute Communication, no. 8. AMS and Fields Institute, Calgary (1996)
2.
3.
Arnol’d, V.I.: Arnold’s Problems. Springer, Berlin (2004). Translated and revised edition of the 2000 Russian original
4.
Arnol’d, V.I., Goussein-Zade, S.M., Varchenko, A.N.: Singularities of Differential Mappings. Birkhauser, Boston (1988)CrossRef
5.
Assémat, E., Efstathiou, K., Joyeux, M., Sugny, D.: Fractional bidromy in the vibrational spectrum of HOCl. Phys. Rev. Lett. 104, 113002 (2010)CrossRef
6.
7.
8.
Binyamini G., Novikov, D., Yakovenko, S.: On the number of zeros of Abelian integrals: a constructive solution of the infinitesimal Hilbert sixteenth problem. Invent. Math. 181, 227 (2010)MathSciNetCrossRefMATH
9.
Bolsinov, A.V., Fomenko, A.T.: Integrable Hamiltonian Systems: Geometry, Topology, Classification. Chapman and Hall/CRC, Boca Raton (2004)MATH
10.
Cushman, R.H., Bates, L.: Global Aspects of Classical Integrable Systems. Birkhäuser, Basel (1997)CrossRefMATH
11.
Cushman, R.H., Dullin, H.R., Giacobbe, A., Holm, D.D., Joyeux, M., Lynch, P., Sadovskii, D.A., Zhilinskii, B.I.: CO2 molecule as a quantum realization of the 1:1:2 resonant swing-spring with monodromy. Phys. Rev. Lett. 93, 024302 (2004)CrossRef
12.
13.
14.
15.
Efstathiou, K.: Metamorphoses of Hamiltonian Systems with Symmetries. Lecture Notes in Mathematics, vol. 1864. Springer, Berlin (2005)
16.
17.
Efstathiou, K., Sadovskií, D.A.: Normalization and global analysis of perturbations of the hydrogen atom. Rev. Mod. Phys. 82, 2099 (2010)CrossRef
18.
19.
Efstathiou, K., Joyeux, M., Sadovskií, D.A.: Global bending quantum number and the absence of monodromy in the HCN-CNH molecule. Phys. Rev. A 69, 032504 (2004)CrossRef
20.
21.
Efstathiou, K., Lukina, O.V., Sadovskií, D.A.: Most typical 1:2 resonant perturbation of the hydrogen atom by weak electric and magnetic fields. Phys. Rev. Lett. 101, 253003 (2008)CrossRef
22.
23.
24.
Grondin, L., Sadovskií, D.A., Zhilinskii, B.I.: Monodromy as topological obstruction to global action-angle variables in systems with coupled angular momenta and rearrangement of bands in quantum spectra. Phys. Rev. A 65, 012105 (2001)CrossRef
25.
26.
Ilyashenko, Y.S.: Appearance of limit cycles by perturbation of the equation \(dw/dz = Rz/Rw\), where R(z, w) is a polynomial. Mat. Sbornik (New Series) 78, 360 (1969)
27.
Khovanskii, A.: Real analytic manifolds with the property of finiteness and complex abelian integrals. Funktsional. Anal. i Prilozhen. 18, 40 (1984)MathSciNetCrossRef
28.
Kirwan, F.: Complex Algebraic Curves. Cambridge University Press, Cambridge (1993)MATH
29.
Llibre, J., Pedregal, P.: Hilbert’s 16th problem. When variational principles meet differential systems (2015). arXiv: 14116814
30.
Nekhoroshev, N.N.: Action-angle variables and their generalizations. Trans. Mosc. Math. Soc. 26, 180 (1972)MathSciNetMATH
31.
Nekhoroshev, N.N., Sadovskií, D.A., Zhilinskií, B.I.: Fractional Hamiltonian monodromy. Ann. Inst. Henri Poincaré 7, 1099 (2006)MathSciNetCrossRefMATH
32.
Sadovskií, D.A., Zhilinskií, B.I.: Hamiltonian systems with detuned 1:1:2 resonance: Manifestation of bidromy. Ann. Phys. 32, 164 (2007)MathSciNetCrossRefMATH
33.
Sugny, D., Mardesic, P., Pelletier, M., Jebrane, A., Jauslin, H.R.: Fractional Hamiltonian monodromy from a Gauss-Manin monodromy. J. Math. Phys. 49, 042701 (2008)MathSciNetCrossRefMATH
34.
Varchenko, A.N.: Estimation of the number of zeros of an abelian integral depending on a parameter and limit cycles. Funct. Anal. Appl. 18, 14 (1984)MathSciNetCrossRefMATH
35.
36.
37.
Winnewisser, B.P., Winnewisser, M., Medvedev, I.R., Behnke, M., De Lucia, F.C., Ross, S.C., Koput, J.: Experimental confirmation of quantum monodromy: the millimeter wave spectrum of cyanogen isothiocyanate NCNCS. Phys. Rev. Lett. 95, 243002 (2005)CrossRef
38.
Zoladek, H.: The Monodromy Group. Birkhauser, Boston (2006)MATH
Metadata
Title
Abelian Integrals: From the Tangential 16th Hilbert Problem to the Spherical Pendulum
Authors
Pavao Mardešić
Dominique Sugny
Léo Van Damme
Copyright Year
2016
Book
Mathematical Sciences with Multidisciplinary Applications

Print ISBN: 978-3-319-31321-4

Electronic ISBN: 978-3-319-31323-8

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-31323-8_15

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