A note on strange nonchaotic attractors

Gerhard Keller

Fundamenta Mathematicae (1996)

  • Volume: 151, Issue: 2, page 139-148
  • ISSN: 0016-2736

Abstract

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For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ,x) ∈ 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 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 T 1 × ℝ + is Γ ̅ , but for a residual set of points in 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.

How to cite

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Keller, Gerhard. "A note on strange nonchaotic attractors." Fundamenta Mathematicae 151.2 (1996): 139-148. <http://eudml.org/doc/212186>.

@article{Keller1996,
abstract = {For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ,x) ∈ $\{T\}^1 × ℝ_+$ 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 $\{T\}^1 ×ℝ_+$. 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 $\{T\}^1 × ℝ_+$ is $Γ̅$, but for a residual set of points in $\{T\}^1 × ℝ_+$ 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.},
author = {Keller, Gerhard},
journal = {Fundamenta Mathematicae},
keywords = {quasiperiodically forced dynamical systems; strange nonchaotic attractors},
language = {eng},
number = {2},
pages = {139-148},
title = {A note on strange nonchaotic attractors},
url = {http://eudml.org/doc/212186},
volume = {151},
year = {1996},
}

TY - JOUR
AU - Keller, Gerhard
TI - A note on strange nonchaotic attractors
JO - Fundamenta Mathematicae
PY - 1996
VL - 151
IS - 2
SP - 139
EP - 148
AB - For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ,x) ∈ ${T}^1 × ℝ_+$ 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 ${T}^1 ×ℝ_+$. 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 ${T}^1 × ℝ_+$ is $Γ̅$, but for a residual set of points in ${T}^1 × ℝ_+$ 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.
LA - eng
KW - quasiperiodically forced dynamical systems; strange nonchaotic attractors
UR - http://eudml.org/doc/212186
ER -

References

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  1. [1] U. Bellack, talk at the Plenarkolloquium des Forschungsschwerpunkts "Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme", July 12, 1995, Bad Windsheim. 
  2. [2] C. Grebogi, E. Ott, S. Pelikan and J. A. Yorke, Strange attractors that are not chaotic, Phys. D 13 (1984), 261-268. Zbl0588.58036
  3. [3] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z. 180 (1982), 119-140. Zbl0485.28016
  4. [4] I. Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. Amer. Math. Soc. 31 (1994), 68-74. Zbl0853.58077
  5. [5] M. St. Pierre, Diplomarbeit, Erlangen, 1994. 
  6. [6] A. S. Pikovsky and U. Feudel, Characterizing strange nonchaotic attractors, Chaos 5 (1995), 253-260. Zbl1055.37519

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