Singularity Analysis of 2D Complexified Hamiltonian Systems

  • S. Abenda
  • A. Bazzani
Part of the NATO ASI Series book series (NSSB, volume 331)


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).


Riemann Surface Hamiltonian System Analytic Continuation Action Variable Angle Variable 
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  1. M. Adler and P. van Moerbeke, 1988, “Algebraic completely integrable systems: a systematic approach,” Perspectives in Mathematics Academic Press, Boston.Google Scholar
  2. V.I. Arnol’d, 1978, “Mathematical Methods of Classical Mechanics,” Springer Verlag.Google Scholar
  3. V.I. Arnol’d, 1983, “Geometrical Methods in the theory of ordinary differential equations,” Springer Verlag.Google Scholar
  4. A. Bazzani and G. Turchetti, 1992, Singularities of normal forms and topology of the orbits in area-preserving maps, J. Phys. A 25:L427.CrossRefzbMATHMathSciNetADSGoogle Scholar
  5. 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.Google Scholar
  6. L. Billi, M. Malvasi and G. Turchetti, 1993, Natural boundaries of normalizing transformations, Contribution to this volume.Google Scholar
  7. 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.CrossRefzbMATHMathSciNetADSGoogle Scholar
  8. 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.CrossRefzbMATHMathSciNetADSGoogle Scholar
  9. A.D. Bruno, 1990, A comparison of conditions on small divisors, IHES M: 36.Google Scholar
  10. V.V. Kozlov, 1983, Integrability and non-integrability in Hamiltonian Mechanics, Russian Math. Surveys 38:1–76.CrossRefzbMATHADSGoogle Scholar
  11. P. van Moerbeke, 1987, Introduction to Algebraic Integrable Systems and their Painlevé analysis, in: “Bowdoin AMS Summer Symposium,”, ed., preprint.Google Scholar
  12. C.L. Siegel, 1971, “Topics in Complex Function Theory, vol. 1,2,3,” Wiley Interscience.Google Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • S. Abenda
    • 1
  • A. Bazzani
    • 1
  1. 1.Department of Mathematics, INFN, Sezione di BolognaUniversity of BolognaBolognaItaly

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