Ukrainian Mathematical Journal

, Volume 55, Issue 1, pp 82–92 | Cite as

Lyapunov–Schmidt Approach to Studying Homoclinic Splitting in Weakly Perturbed Lagrangian and Hamiltonian Systems

  • A. M. Samoilenko
  • A. K. Prykarpats'kyi
  • V. H. Samoilenko


We analyze the geometric structure of the Lyapunov–Schmidt approach to studying critical manifolds of weakly perturbed Lagrangian and Hamiltonian systems.


Hamiltonian System Geometric Structure Critical Manifold Weakly Perturb Schmidt Approach 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Wiggings, “Global bifurcation and chaos,” Appl. Math. Sci., 73, 370 (1998).Google Scholar
  2. 2.
    L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Inst. Math., New York (1974).Google Scholar
  3. 3.
    J. K. Hale, Ordinary Differential Equations, Wiley (1980).Google Scholar
  4. 4.
    A. M. Samoilenko, O. Ya. Tymchyshyn, and A. K. Prykarpats'kyi, “The Poincaré - Mel'nikov geometric analysis of the transversal splitting of separatrix manifolds for slowly perturbed nonlinear dynamical systems,” Ukr. Mat. Zh., 45, No. 12, 1668–1681 (1993).Google Scholar
  5. 5.
    V. K. Mel'nikov, “On the center stability under periodic perturbations,” Proc. Moscow Math. Soc., 2, No. 1, 3–52 (1963).Google Scholar
  6. 6.
    V. I. Arnol'd, Mathematical Methods in Classical Mechanics [in Russian], Nauka, Moscow (1989).Google Scholar
  7. 7.
    H. Poincaré, New Methods of Celestial Mechanics, Vols. 1-3, Hermann, Paris (1912).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • A. M. Samoilenko
    • 1
  • A. K. Prykarpats'kyi
    • 2
  • V. H. Samoilenko
    • 3
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKiev
  2. 2.Institute for Applied Problems in Mechanics and Mathematics, Ukrainian Academy of SciencesLviv; University of Mining and MetallurgyPoland
  3. 3.Shevchenko Kiev UniversityKiev

Personalised recommendations