[Une application de type Hénon cubique dans le déploiement d'une orbite homocline dégénérée avec résonance]
Nous étudions le déploiement d'un champ de vecteurs sur qui possède une orbite homocline dégénérée associée à une singularité hyperbolique. La partie linéaire du champ en cette singulartité posssède une résonance mais, pour le système initial, le terme résonant associé à cette résonance disparaît. Nous montrons qu'après changement d'échelle, l'application de retour de Poincaré sur une section transverse est proche d' une application de Hénon cubique. Un attracteur étrange est présent et persiste au sens de la mesure de Lebesgue. Nous montrons également la présence d'un attracteur avec une entropie topologique proche de .
In this Note, we study the unfolding of a vector field that possesses a degenerate homoclinic (of inclination-flip type) to a hyperbolic equilibrium point where its linear part possesses a resonance. For the unperturbed system, the resonant term associated with the resonance vanishes. After suitable rescaling, the Poincaré return map is a cubic Hénon-like map. We deduce the existence of a strange attractor which persists in the Lebesgue measure sense. We also show the presence of an attractor with topological entropy close to .
Accepté le :
Publié le :
@article{CRMATH_2005__340_11_843_0,
author = {Martens, Marco and Naudot, Vincent and Yang, Jiazhong},
title = {A cubic {H\'enon-like} map in the unfolding of degenerate homoclinic orbit with resonance},
journal = {Comptes Rendus. Math\'ematique},
pages = {843--846},
publisher = {Elsevier},
volume = {340},
number = {11},
year = {2005},
doi = {10.1016/j.crma.2005.04.001},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2005.04.001/}
}
TY - JOUR AU - Martens, Marco AU - Naudot, Vincent AU - Yang, Jiazhong TI - A cubic Hénon-like map in the unfolding of degenerate homoclinic orbit with resonance JO - Comptes Rendus. Mathématique PY - 2005 SP - 843 EP - 846 VL - 340 IS - 11 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2005.04.001/ DO - 10.1016/j.crma.2005.04.001 LA - en ID - CRMATH_2005__340_11_843_0 ER -
%0 Journal Article %A Martens, Marco %A Naudot, Vincent %A Yang, Jiazhong %T A cubic Hénon-like map in the unfolding of degenerate homoclinic orbit with resonance %J Comptes Rendus. Mathématique %D 2005 %P 843-846 %V 340 %N 11 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2005.04.001/ %R 10.1016/j.crma.2005.04.001 %G en %F CRMATH_2005__340_11_843_0
Martens, Marco; Naudot, Vincent; Yang, Jiazhong. A cubic Hénon-like map in the unfolding of degenerate homoclinic orbit with resonance. Comptes Rendus. Mathématique, Tome 340 (2005) no. 11, pp. 843-846. doi : 10.1016/j.crma.2005.04.001. http://www.numdam.org/articles/10.1016/j.crma.2005.04.001/
[1] Asymptotic properties of the Dulac map near a hyperbolic saddle in dimension three, Ann. Fac. Sci. Toulouse Math. (8), Volume 6 (2001) no. 4, pp. 595-617
[2] Linearization of germs of hyperbolic vector fields, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003), pp. 19-22
[3] Homoclinic bifurcation at resonant eigenvalues, J. Dynamics Differential Equations, Volume 2 (1990), pp. 177-244
[4] Homoclinic twisting bifurcation cusp horseshoe maps, J. Dynamics Differential Equations, Volume 5 (1993), pp. 417-467
[5] Strange attractors in saddle cycles: prevalence and globality, Invent. Math., Volume 125 (1996), pp. 37-74
[6] Double impulse solutions in nerve axon equations, SIAM J. Appl. Math., Volume 42 (1982), pp. 219-234
[7] Invariant Manifolds, Lecture Notes in Math., vol. 583, Springer, 1977
[8] A non linear oscillator with a strange attractor, Philos. Trans. Roy. Soc. London Ser. A, Volume 292 (1979), pp. 419-448
[9] Bifurcations to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis, Automatica, Volume 14 (1978), pp. 367-384
[10] Global aspects of homoclinic bifurcations of vector fields, Mem. Amer. Math. Soc., Volume 121 (1996)
[11] The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit, Ergodic Theory Dynamical Systems, Volume 14 (1994), pp. 667-693
[12] Homoclinic doubling cascade, Arch. Rational Mech. Anal., Volume 160 (2001), pp. 195-243
[13] Resonant homoclinic flip bifurcations, J. Dynamics Differential Equations, Volume 12 (2000) no. 4, pp. 807-850
[14] The existence of infinitely many homoclinic doubling bifurcations from some codimension 3 homoclinic orbits, J. Dynamics Differential Equations, Volume 9 (1997) no. 3, pp. 445-462
[15] Strange attractor in the unfolding of an inclination-flip homoclinic orbit, Ergodic Theory Dynamical Systems, Volume 16 (1996), pp. 1071-1086
[16] A strange attractor in the unfolding of an orbit-flip homoclinic orbit, Dynamical Systems: An International Journal, Volume 17 (2002) no. 1, pp. 45-63
[17] V. Naudot, J. Yang, Linearization of families of germs, Preprint, 2003
[18] Homoclinic bifurcation to a transitive attractor of Lorenz type, Nonlinearity, Volume 2 (1989), pp. 495-518
[19] Lorenz attractors through Shil'nikov-type bifurcation. Part I, Ergodic Theory Dynamical Systems, Volume 10 (1990), pp. 793-821
[20] B. Sandstede, Verzweigungstheorie homokliner Verdopplungen, PhD thesis, University of Stuttgart, 1993
[21] A case of the existence of a countable number of periodic motions, Soviet Math. Dokl., Volume 6 (1965), pp. 163-166
Cité par Sources :
