Abstract
The connection between the integrability of a differential equation and the nature of the singularities in the complex time of the solutions, has been studied by various authors since Painlevè (Bountis et. al. 1991). Recently it has been considered the connection of the integrability criterion of Painlevè and the algebraic integrability of Arnol’d-Liouville (Arnol’d 1978) for the Hamiltonian systems (M. Adler and P. van Moerbeke 1988, P. van Moerbeke 1987); unfortunately the family of algebraic integrable systems is very poor with respect to integrable systems according to Arnol’d-Liouville. On the other hand some non-integrability criteria have been established for the Hamiltonian systems (V.V. Kozlov 1983).
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References
M. Adler and P. van Moerbeke, 1988, “Algebraic completely integrable systems: a systematic approach,” Perspectives in Mathematics Academic Press, Boston.
V.I. Arnol’d, 1978, “Mathematical Methods of Classical Mechanics,” Springer Verlag.
V.I. Arnol’d, 1983, “Geometrical Methods in the theory of ordinary differential equations,” Springer Verlag.
A. Bazzani and G. Turchetti, 1992, Singularities of normal forms and topology of the orbits in area-preserving maps, J. Phys. A 25:L427.
A. Bazzani and G. Turchetti, 1993, Analyticity of normalizing transformations for area-preserving maps, in: “Chaotic Dynamics: Theory and Practise,” T. Bountis, ed., Springer Verlag.
L. Billi, M. Malvasi and G. Turchetti, 1993, Natural boundaries of normalizing transformations, Contribution to this volume.
T. Bountis, V. Papageorgiou and M. Bier, 1987, On the singularity analysis of intersecting separatrices in near-integrable dynamical systems, Physica D 24:292–304.
T. Bountis, L. Drossos and I.C. Percival, 1991, Non-integrable systems with algebraic singularities in complex time, J.Phys. A Gen. 24:3217–3236.
A.D. Bruno, 1990, A comparison of conditions on small divisors, IHES M: 36.
V.V. Kozlov, 1983, Integrability and non-integrability in Hamiltonian Mechanics, Russian Math. Surveys 38:1–76.
P. van Moerbeke, 1987, Introduction to Algebraic Integrable Systems and their Painlevé analysis, in: “Bowdoin AMS Summer Symposium,”, ed., preprint.
C.L. Siegel, 1971, “Topics in Complex Function Theory, vol. 1,2,3,” Wiley Interscience.
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© 1994 Springer Science+Business Media New York
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Abenda, S., Bazzani, A. (1994). Singularity Analysis of 2D Complexified Hamiltonian Systems. In: Seimenis, J. (eds) Hamiltonian Mechanics. NATO ASI Series, vol 331. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-0964-0_18
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DOI: https://doi.org/10.1007/978-1-4899-0964-0_18
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