Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium

Massimiliano Berti; Philippe Bolle

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1998)

  • Volume: 9, Issue: 3, page 167-175
  • ISSN: 1120-6330

Abstract

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We consider autonomous Lagrangian systems possessing two homoclinic orbits to an hyperbolic equilibrium of saddle-saddle type with two different characteristic exponents. Under a nondegeneracy assumption on the homoclinics and under suitable conditions on the geometric behaviour of these homoclinics near the equilibrium we show, by variational methods, that they give rise to an infinite family of multibump homoclinic solutions. We relax the nondegeneracy assumption when the two characteristic exponents are close one to the other.

How to cite

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Berti, Massimiliano, and Bolle, Philippe. "Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 9.3 (1998): 167-175. <http://eudml.org/doc/252321>.

@article{Berti1998,
abstract = {We consider autonomous Lagrangian systems possessing two homoclinic orbits to an hyperbolic equilibrium of saddle-saddle type with two different characteristic exponents. Under a nondegeneracy assumption on the homoclinics and under suitable conditions on the geometric behaviour of these homoclinics near the equilibrium we show, by variational methods, that they give rise to an infinite family of multibump homoclinic solutions. We relax the nondegeneracy assumption when the two characteristic exponents are close one to the other.},
author = {Berti, Massimiliano, Bolle, Philippe},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Homoclinic orbits; Chaos; Saddle-saddle equilibrium; Variational methods; autonomous Lagrangian systems; chaotic behaviour; zero energy level; homoclinic orbits; multibump homoclinic solutions},
language = {eng},
month = {9},
number = {3},
pages = {167-175},
publisher = {Accademia Nazionale dei Lincei},
title = {Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium},
url = {http://eudml.org/doc/252321},
volume = {9},
year = {1998},
}

TY - JOUR
AU - Berti, Massimiliano
AU - Bolle, Philippe
TI - Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1998/9//
PB - Accademia Nazionale dei Lincei
VL - 9
IS - 3
SP - 167
EP - 175
AB - We consider autonomous Lagrangian systems possessing two homoclinic orbits to an hyperbolic equilibrium of saddle-saddle type with two different characteristic exponents. Under a nondegeneracy assumption on the homoclinics and under suitable conditions on the geometric behaviour of these homoclinics near the equilibrium we show, by variational methods, that they give rise to an infinite family of multibump homoclinic solutions. We relax the nondegeneracy assumption when the two characteristic exponents are close one to the other.
LA - eng
KW - Homoclinic orbits; Chaos; Saddle-saddle equilibrium; Variational methods; autonomous Lagrangian systems; chaotic behaviour; zero energy level; homoclinic orbits; multibump homoclinic solutions
UR - http://eudml.org/doc/252321
ER -

References

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  1. Berti, M. - Bolle, P., Homoclinics and Chaotic Behaviour for Perturbed Second Order Systems. Preprint n. 3, febbraio 1997, Scuola Normale Superiore, Pisa; Ann. di Mat. Pura e Applicata, to appear. Zbl0957.37019MR1746547DOI10.1007/BF02506001
  2. Berti, M. - Bolle, P., Variational construction of Homoclinics and Chaotic Behaviour in presence of a saddle-saddle equilibrium. Preprint n. 11, aprile 1998, Scuola Normale Superiore, Pisa. Zbl0938.34039
  3. Bolotin, S. - Rabinowitz, P., A variational construction of chaotic trajectories for a Hamiltonian system on a torus. Bull. Union. Math. Ital., 1997, to appear. Zbl0957.70020MR1662325
  4. Buffoni, B. - Séré, E., A global condition for quasi-random behaviour in a class of conservative systems. Comm. in Pure and Appl. Math., 1996. Zbl0860.58027
  5. Hofer, H., A geometric description of the neighborhood of a critical point given by the mountain pass theorem. T. London Math. Soc., (2) 31, 1985, 566-570. Zbl0573.58007MR812787DOI10.1112/jlms/s2-31.3.566
  6. Holmes, P. J., Periodic, non-periodic and irregular motions in a Hamiltonian system. Rocky Mountain J. Math., 10, 1980, 679-693. Zbl0427.70026MR595097DOI10.1216/RMJ-1980-10-4-679
  7. Turaev, D. V. - Shil'nikov, L. P., On Hamiltonian systems with homoclinic curves of a saddle. Dokl. AN SSSR, 304, 1989, 811-814. Zbl0689.58013MR988994
  8. Wiggins, S., Global Bifurcation and Chaos. Applied Mathematical Sciences, vol. 73, Springer-Verlag, 1988. Zbl0661.58001MR956468DOI10.1007/978-1-4612-1042-9

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