On the Hausdorff dimension of fractal attractors

@article{Grassberger1981OnTH,
  title={On the Hausdorff dimension of fractal attractors},
  author={Peter Grassberger},
  journal={Journal of Statistical Physics},
  year={1981},
  volume={26},
  pages={173-179}
}
  • P. Grassberger
  • Published 1 September 1981
  • Mathematics
  • Journal of Statistical Physics
We consider such mappingsxn+1=F(xn) of an interval into itself for which the attractor is a Cantor set. For the same class of mappings for which the Feigenbaum scaling laws hold, we show that the Hausdorff dimension of the attractor is universally equal toD=0.538 ... 
View on Springer
101 Citations
Highly Influential Citations
7
Background Citations
12
Methods Citations
10

101 Citations

Citation Type

Citation Type

On hausdorff dimension of some cantor attractors
We study what happens with the dimension of Feigenbaum-like attractors of smooth unimodal maps as the order of the critical point grows.
Fractal dimension of the strange attractor in a piecewise linear two-dimensional map
Abstract We calculate the fractal dimension of the strange attractor in the map x ′ = ax - sgn ( x ) + bz , z ′ = x . The method is based on the construction of the unstable manifolds of period-two…
Fractal dimensions and homeomorphic conjugacies
We investigate the behavior of the spectrum of singularities associated with the invariant measure of some dynamical systems under nonsmooth coordinate changes. When the homeomorphic conjugacy is not…
Dimension of invariant measures for maps with exponent zero
We give examples of maps of the interval with zero entropy for which the continuous invariant measure has no dimension, and we prove a dimension property for maps lying in the stable manifold of…
On the metric properties of the Feigenbaum attractor
The two standard literature definitions of the function associated with the Feigenbaum attractor are not equivalent. The method due to Vulet al. and Feigenbaum is used to calculate the Haussdorff…
Fractal dimension and degeneracy of the critical point for iterated maps
Variations of the three most commonly used definitions of fractal dimension, namely the capacity, the information dimension and the correlation exponent, with the degeneracy of the critical point is…
The geometry of fractal sets
Preface Introduction Notation 1. Measure and dimension 2. Basic density properties 3. Structure of sets of integral dimension 4. Structure of sets of non-integral dimension 5. Comparable net measures…
Classifying and quantifying basins of attraction.
TLDR
A scheme is proposed to classify the basins for attractors of dynamical systems in arbitrary dimensions using a Monte Carlo method and is applied to numerous common dissipative chaotic maps and flows in various dimensions.
Rigorous bounds on the Hausdorff dimension of Feigenbaum attractors
We calculate rigorous bounds on the Hausdorff dimension of the attractor at the accumulation of the perioddoubling cascade for families of maps with quadratic, cubic, and quartic critical point. To…
Cantor Set as a Fractal and Its Application in Detecting Chaotic Nature of Piecewise Linear Maps
We have investigated the Cantor set from the perspective of fractals and box-counting dimension. Cantor sets can be constructed geometrically by continuous removal of a portion of the closed unit…
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 11 REFERENCES
Dimension of Strange Attractors
A relationship between the Lyapunov numbers of a map with a strange attractor and the dimension of the strange attractor has recently been conjectured. Here, the conjecture is numerically tested with…
Universal properties of maps on an interval
We consider itcrates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied.…
The universal metric properties of nonlinear transformations
AbstractThe role of functional equations to describe the exact local structure of highly bifurcated attractors ofxn+1 =λf(xn) independent of a specificf is formally developed. A hierarchy of…
Fractals: Form, Chance and Dimension
This is the most extraordinarily beautiful book in thought and in form that I have read for many years, and that is all the more peculiar for its being a somewhat technically mathematical treatise.…
Quantitative universality for a class of nonlinear transformations
AbstractA large class of recursion relationsxn + 1 = λf(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The…
Sequences of infinite bifurcations and turbulence in a five-mode truncation of the Navier-Stokes equations
Two infinite sequences of orbits leading to turbulence in a five-mode truncation of the Navier-Stokes equations for a 2-dimensional incompressible fluid on a torus are studied in detail. Their…
Bifurcations and Dynamic Complexity in Simple Ecological Models
TLDR
It is shown that as a hump steepens, the dynamics goes from a stable point, to a bifurcating hierarchy of stable cycles of period 2n, into a region of chaotic behavior where the population exhibits an apparently random sequence of "outbreaks" followed by "crashes".
Rayleigh-Bénard experiment in liquid helium ; frequency locking and the onset of turbulence
2014 In a Rayleigh-Bénard experiment in liquid helium, we study the time-dependent regimes, above the onset of convection, in a parallelepipedic cell with two convective rolls of oblate cross-section…
On the nature of turbulence
A mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed.
Fractal Dimensions of Chaotic Flows of Autonomous Dissipative Systems
...
1
2
...