# A note on strange nonchaotic attractors

Fundamenta Mathematicae (1996)

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

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topKeller, 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

top- [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. Zbl0588.58036
- [3] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z. 180 (1982), 119-140. Zbl0485.28016
- [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] M. St. Pierre, Diplomarbeit, Erlangen, 1994.
- [6] A. S. Pikovsky and U. Feudel, Characterizing strange nonchaotic attractors, Chaos 5 (1995), 253-260. Zbl1055.37519

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