ArticleOriginal scientific text

Title

A note on strange nonchaotic attractors

Authors 1

Affiliations

  1. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, D-91054 Erlangen, Germany

Abstract

For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ,x) ∈ {symT}1×+ with nonpositive Lyapunov exponents we prove the existence of an attractor Γ̅ with the following properties:  1. Γ̅ is the closure of the graph of a function x = ϕ(θ). It attracts Lebesgue-a.e. starting point in {symT}1×+. The set {θ:ϕ(θ) ≠ 0} is meager but has full 1-dimensional Lebesgue measure.  2. The omega-limit of Lebesgue-a.e point in {symT}1×+ is Γ̅, but for a residual set of points in {symT}1×+ the omega limit is the circle {(θ,x):x = 0} contained in Γ̅.  3. Γ̅ is the topological support of a BRS measure. The corresponding measure theoretical dynamical system is isomorphic to the forcing rotation.

Bibliography

  1. U. Bellack, talk at the Plenarkolloquium des Forschungsschwerpunkts "Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme", July 12, 1995, Bad Windsheim.
  2. C. Grebogi, E. Ott, S. Pelikan and J. A. Yorke, Strange attractors that are not chaotic, Phys. D 13 (1984), 261-268.
  3. F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z. 180 (1982), 119-140.
  4. I. Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. Amer. Math. Soc. 31 (1994), 68-74.
  5. M. St. Pierre, Diplomarbeit, Erlangen, 1994.
  6. A. S. Pikovsky and U. Feudel, Characterizing strange nonchaotic attractors, Chaos 5 (1995), 253-260.
Pages:
139-148
Main language of publication
English
Received
1995-10-10
Accepted
1996-03-06
Published
1996
Exact and natural sciences