Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Singular hyperbolic systems

Authors: C. A. Morales, M. J. Pacifico and E. R. Pujals
Journal: Proc. Amer. Math. Soc. 127 (1999), 3393-3401
MSC (1991): Primary 58F10, 58F15
Published electronically: May 4, 1999
MathSciNet review: 1610761
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We construct a class of vector fields on 3-manifolds containing the hyperbolic ones and the geometric Lorenz attractor. Conversely, we shall prove that nonhyperbolic systems in this class resemble the Lorenz attractor: they have Lorenz-like singularities accumulated by periodic orbits and they cannot be approximated by flows with nonhyperbolic critical elements.

References [Enhancements On Off] (What's this?)

  • [1] Rodrigo Bamón, Rafael Labarca, Ricardo Mañé, and María José Pacífico, The explosion of singular cycles, Inst. Hautes Études Sci. Publ. Math. 78 (1993), 207–232 (1994). MR 1259432
  • [2] Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
  • [3] Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541
  • [4] C. I. Doering, Persistently transitive vector fields on three-dimensional manifolds, Dynamical systems and bifurcation theory (Rio de Janeiro, 1985) Pitman Res. Notes Math. Ser., vol. 160, Longman Sci. Tech., Harlow, 1987, pp. 59–89. MR 907891
  • [5] J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale; Applied Mathematical Sciences, Vol. 19. MR 0494309
  • [6] John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582
  • [7] Shuhei Hayashi, Diffeomorphisms in ℱ¹(ℳ) satisfy Axiom A, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 233–253. MR 1176621, 10.1017/S0143385700006726
  • [8] Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR 0271991
  • [9] R. Labarca and M. J. Pacífico, Stability of singularity horseshoes, Topology 25 (1986), no. 3, 337–352. MR 842429, 10.1016/0040-9383(86)90048-0
  • [10] C.A. Morales, M.J. Pacifico, E.R. Pujals, On $C^{1}$-robust transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris 326 (1998), 81-86. CMP 99:02
  • [11] Jacob Palis and Floris Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, 1993. Fractal dimensions and infinitely many attractors. MR 1237641
  • [12] L.P. Shilnikov, Talk at Nizhny-Novgorov University (1996).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 58F10, 58F15

Retrieve articles in all journals with MSC (1991): 58F10, 58F15

Additional Information

C. A. Morales
Affiliation: Université de Bourgogne, Laboratoire de Topologie, B.P.400, 21011, Dijon Cedex-France
Address at time of publication: Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C.P. 68.530, CEP 21.945-970, Rio de Janeiro, Brazil

M. J. Pacifico
Affiliation: Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de Janeiro, Brazil

E. R. Pujals

Keywords: Lorenz attractor, hyperbolicity, Axiom A
Received by editor(s): November 24, 1997
Received by editor(s) in revised form: January 22, 1998
Published electronically: May 4, 1999
Additional Notes: This work was partially supported by CNPq-Brasil, Faperj-Brasil, Pronex-Brasil. The first author was partially supported by CNRS-France.
Communicated by: Mary Rees
Article copyright: © Copyright 1999 American Mathematical Society