Poincaré-Melnikov theory for n-dimensional diffeomorphisms

M. Baldomà; E. Fontich

Applicationes Mathematicae (1998)

  • Volume: 25, Issue: 2, page 129-152
  • ISSN: 1233-7234

Abstract

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We consider perturbations of n-dimensional maps having homo-heteroclinic connections of compact normally hyperbolic invariant manifolds. We justify the applicability of the Poincaré-Melnikov method by following a geometric approach. Several examples are included.

How to cite

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Baldomà, M., and Fontich, E.. "Poincaré-Melnikov theory for n-dimensional diffeomorphisms." Applicationes Mathematicae 25.2 (1998): 129-152. <http://eudml.org/doc/219197>.

@article{Baldomà1998,
abstract = {We consider perturbations of n-dimensional maps having homo-heteroclinic connections of compact normally hyperbolic invariant manifolds. We justify the applicability of the Poincaré-Melnikov method by following a geometric approach. Several examples are included.},
author = {Baldomà, M., Fontich, E.},
journal = {Applicationes Mathematicae},
keywords = {Melnikov function; splitting of separatrices; homoclinic solutions; -dimensional maps; Melnikov functions},
language = {eng},
number = {2},
pages = {129-152},
title = {Poincaré-Melnikov theory for n-dimensional diffeomorphisms},
url = {http://eudml.org/doc/219197},
volume = {25},
year = {1998},
}

TY - JOUR
AU - Baldomà, M.
AU - Fontich, E.
TI - Poincaré-Melnikov theory for n-dimensional diffeomorphisms
JO - Applicationes Mathematicae
PY - 1998
VL - 25
IS - 2
SP - 129
EP - 152
AB - We consider perturbations of n-dimensional maps having homo-heteroclinic connections of compact normally hyperbolic invariant manifolds. We justify the applicability of the Poincaré-Melnikov method by following a geometric approach. Several examples are included.
LA - eng
KW - Melnikov function; splitting of separatrices; homoclinic solutions; -dimensional maps; Melnikov functions
UR - http://eudml.org/doc/219197
ER -

References

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  1. [1] N. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. 
  2. [2] V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Soviet Math. Dokl. 5 (1964), 581-585. Zbl0135.42602
  3. [3] T. Bountis, A. Goriely and M. Kollmann, A Melnikov vector for N-dimensional mappings, Phys. Lett. A 206 (1995), 38-48. Zbl1020.37506
  4. [4] S. N. Chow, J. K. Hale and J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations 37 (1980), 351-373. Zbl0439.34035
  5. [5] A. Delshams and R. Ramírez-Ros, Poincaré-Melnikov-Arnold method for analytic planar maps, Nonlinearity 9 (1996), 1-26. 
  6. [6] A. Delshams and R. Ramírez-Ros, Melnikov potential for exact symplectic maps, preprint, 1996. 
  7. [7] R. W. Easton, Computing the dependence on a parameter of a family of unstable manifolds: generalized Melnikov formulas, Nonlinear Anal. 8 (1984), 1-4. Zbl0536.58027
  8. [8] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971), 193-226. Zbl0246.58015
  9. [9] J. M. Gambaudo, Perturbation de l’application temps τ’ d’un champ de vecteurs intégrable de R 2 , C. R. Acad. Sci. Paris 297 (1987), 245-248. Zbl0546.34035
  10. [10] M. Glasser, V. G. Papageorgiu and T. C. Bountis, Melnikov's function for two-dimensional mappings, SIAM J. Appl. Math. 49 (1989), 692-703. 
  11. [11] J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Sys- tems, and Bifurcations of Vector Fields, Springer, New York, 1983. Zbl0515.34001
  12. [12] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer, New York, 1977. 
  13. [13] H. E. Lomelí, Transversal heteroclinic orbits for perturbed billiards, preprint, 1994. 
  14. [14] V. K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc. 12 (1963), 3-56. 
  15. [15] H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Paris, 1882-1899. 
  16. [16] J. H. Sun, Transversal homoclinic points for high-dimensional maps, preprint, 1994. 
  17. [17] S. Wiggins, Global Bifurcations and Chaos: Analytical Methods, Springer, New York, 1988. Zbl0661.58001

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