Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-07-06T07:43:10.671Z Has data issue: false hasContentIssue false
  • English
  • Français

New Developments in Dynamics: Hyperbolicity and Chaotic Dynamics

Published online by Cambridge University Press:  07 August 2017

J. Palis
J. Palis*
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jd. Botânico, 22.460 - Rio de Janeiro, Brazil

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Two important theories in Dynamical Systems were constructed in the sixties: the hyperbolic theory for general systems and the KAM (after Kolmogorov, Arnold and Moser) theory for conservative systems as the ones that appear in Celestial Mechanics. Most of our discussions here concern dissipative (or locally dissipative) systems, although most questions are now being posed for area preserving maps (symplectic maps in higher dimensions). Moreover, one can argue that understanding dynamically small dissipative perturbations of conservative systems is of much importance: indeed it has been recently shown that a KAM curve (tori in higher dimension) can be destroyed and in fact engulfed in the basin of attraction of a Hénon-like strange attractor as defined below.

Type
Part VII - Dynamical Systems. Maps. Integrators
Information
Symposium - International Astronomical Union , Volume 152: Chaos, Resonance and Collective Dynamical Phenomena in the Solar System , 1992 , pp. 363 - 368
Copyright
Copyright © Kluwer 1992 

References

Araujo, A. and Mañé, R.: 1991 “On the existence of hyperbolic attractors and homoclinic tangencies for surface diffeomorphisms”, to appear.Google Scholar
Benedicks, M. and Carleson, L.: 1991 “The dynamics of the Hénon map”, Annals of Math. 133, 73–169.Google Scholar
Bamón, R., Labarca, R., Mañé, R. and Pacífico, M.J.: 1991, “Bifurcating 3-dimensional singular cycle”, to appear.Google Scholar
Diaz, L. “Persistence of nonhyperbolicity and heterodimensional cycles”, Thesis, IMPA, and to appear.Google Scholar
Diaz, L. Rocha, R. and Viana, M.: 1991, “Saddle-node critical cycles and prevalence of strange attractors”, to appear.Google Scholar
Gambaudo, J.-M., von Strein, S. and Tresser, C.: 1989 “Hénon-like maps with strange attractors: there exist C $iF Kupka-Smale diffeomorphisms on S 2 with neither sinks or sources”, Nonlinearity 2, 287–304.Google Scholar
Guckenheimer, J. and Williams, R.F.:1979 “Structural stability of Lorenz attractors”, Publ. Math. I.H.E.S. 50, 59–72.Google Scholar
Lorenz, E. N.: 1963 “Deterministic non-periodic flow”, J. Atmos. Sci. 20, 130–141.Google Scholar
Labarca, R. and Pací fico, M.J.: 1986 “Stability of singular horseshoe”, Topology 25, 337–352.Google Scholar
Mañé, R. “Contribution to the stability conjecture”, Topology 17, 383–396.Google Scholar
Mora, L. and Viana, M.: 1991 “Abundance of strange attractors”, Acta Math. , to appear.Google Scholar
Newhouse, S.: 1979 “The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms”, Publ. Math. I.H.E.S. 50, 101–151.Google Scholar
Newhouse, S., Palis, J. and Takens, F.: 1983 “Bifurcations and stability of families of diffeomorphisms”, Publ. Math. I.H.E.S. 57, 5–71.CrossRefGoogle Scholar
Palis, J. and Takens, F.: 1987 “Hyperbolicity and the creation of homoclinic orbits”, Annals of Math. 125, 337–374.Google Scholar
Palis, J. and Takens, F.: 1992 Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations , Cambridge University Press.Google Scholar
Palis, J. and Viana, M:1991 “Infinitely many sinks in higher dimensions”, to appear.Google Scholar
Palis, J. and Yoccoz, J.C.: 1989 “Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension”, to appear.Google Scholar
Robinson, C.: 1989 “Homoclinic bifurcation to a transitive attractor of Lorenz type”, Nonlinearity 2, 495–518.CrossRefGoogle Scholar
Rovella, A.: 1991 “The dynamics of the perturbations of the contracting Lorenz attractor”, Thesis, IMPA, and to appear.Google Scholar
Rychlik, M.:1990 “Lorenz's attractors through Silnikov-type bifurcation”, Erg. Th. and Dyn. Syst. 10, 793–821.Google Scholar
Silnikov, L.P.:1965 “A case of the existence of denumerable set of periodic motions”, Sov. Math. Dokl. 6, 163–166.Google Scholar
Takens, F.: 1991 “On the geometry of non-transversal intersections of invariant manifolds and scaling properties of bifurcation sets”, Pitman Research Notes in Math. Series , to appear.Google Scholar
Takens, F.: 1986 “Homoclinic bifurcations” Proc. Int. Congress of Math., Berkeley, 1229–1236.Google Scholar
Viana, M.: 1991 “Strange attractors in higher dimensions”, Thesis, IMPA, and to appear.Google Scholar
Williams, R.F.: 1979 “The structure of Lorenz attractors”, Publ. Math. I.H.E.S. 50, 101–152.CrossRefGoogle Scholar
Yorke, J.A. and Alligood, K. T.: 1983 “Cascades of period doubling bifurcations: a prerequisite for horseshoes”, Bull. A.M.S. 9, 319–322.Google Scholar