A note on strange nonchaotic attractors

• Autorzy: Gerhard Keller
• Języki publikacji: EN

Abstrakty

EN

For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ,x) ∈ ${\sym T}^1 × ℝ_+$ 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 ${\sym T}^1 ×ℝ_+$. 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 ${\sym T}^1 × ℝ_+$ is $Γ̅$, but for a residual set of points in ${\sym T}^1 × ℝ_+$ 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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1996
• Tom: 151
• Numer: 2
• Strony: 139-148

Daty

wydano

1996

otrzymano

1995-10-10

poprawiono

1996-03-06

poprawiono

1996-06-24

Twórcy

autor

Gerhard Keller

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

Bibliografia

• [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.