Dimensions of strange nonchaotic attractors (original) (raw)
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Is the dimension of chaotic attractors invariant under coordinate changes?
Journal of Statistical Physics, 1984
Several different dimensionlike quantities, which have been suggested as being relevant to the study of chaotic attractors, are examined. In particular, we discuss whether these quantities are invariant under changes of variables that are differentiable except at a finite number of points. It is found that some are and some are not. It is suggested that the word "dimension" be reversed only for those quantities have this invariance property.
D S ] 1 3 A pr 2 00 6 How chaotic are strange nonchaotic attractors ?
2014
We show that the classic examples of quasi-periodically forced maps with strange nonchaotic attractors described by Grebogi et al and Herman in the mid-1980s have some chaotic properties. More precisely, we show that these systems exhibit sensitive dependence on initial conditions, both on the whole phase space and restricted to the attractor. The results also remain valid in more general classes of quasiperiodically forced systems. Further, we include an elementary proof of a classic result by Glasner and Weiss on sensitive dependence, and we clarify the structure of the attractor in an example with two-dimensional fibers also introduced by Grebogi et al.
Dimension of strange attractors in four-dimensional maps
Physics Letters A, 1990
The method of Tsang for calculating the Hausdorifdimension ofa strange attractor in two-dimensional maps is generalized to a four-dimensional map for the double rotor system. The computed dimension compares well with previous work, and is much faster computationally.
Fractal Properties of Robust Strange Nonchaotic Attractors
Physical Review Letters, 2001
We consider the existence of robust strange nonchaotic attractors (SNA's) in a simple class of quasiperiodically forced systems. Rigorous results are presented demonstrating that the resulting attractors are strange in the sense that their box-counting dimension is N +1 while their information dimension is N. We also show how these properties are manifested in numerical experiments.
2011
The correlation dimension of the fractal attractor of the general two-dimensional iterated quadratic map is obtained by using the cell-count algorithm. However, the cell-size necessary to cover all points of the attractor object is determined. Also, an explicit criterion for the selection of the cell size in cell-count algorithm is presented. When the log-log graph is plotted by Matlab program, the result will be an approximate polynomial of degree one (straight line) that fits the data in a least squares method. The correlation dimension of the fractal attractor object is the slope of this straight line. Mathematics Subject Classification: 28A80
Unstable periodic orbits and the dimensions of multifractal chaotic attractors
Physical review. A, 1988
The probability measure generated by typical chaotic orbits of a dynamical system can have an arbitrarily Sne-scaled interwoven structure of points with different singularity scalings. Recent work has characterized such measures via a spectrum of fractal dimension values. In this paper we pursue the idea that the infinite number of unstable periodic orbits embedded in the support of the measure provides the key to an understanding of the structure of the subsets with different singularity scalings. In particular, a formulation relating the spectrum of dimensions to unstable periodic orbits is presented for hyperbolic maps of arbitrary dimensionality. Both chaotic attractors and chaotic repeBers are considered.
Unstable dimension variability: A source of nonhyperbolicity in chaotic systems
International Symposium on Physical Design, 1996
The hyperbolicity or nonhyperbolicity of a chaotic set has profound implications for the dynamics on the set. A familiar mechanism causing nonhyperbolicity is the tangency of the stable and unstable manifolds at points on the chaotic set. Here we investigate a different mechanism that can lead to nonhyperbolicity in typical invertible (respectively noninvertible) maps of dimension 3 (respectively 2) and higher. In particular, we investigate a situation (first considered by Abraham and Smale in 1970 for different purposes) in which the dimension of the unstable (and stable) tangent spaces are not constant over the chaotic set; we call this unstable dimension variability. A simple two-dimensional map that displays behavior typical of this phenomenon is presented and analyzed.
How chaotic are strange non-chaotic attractors?
Nonlinearity, 2006
We show that the classic examples of quasi-periodically forced maps with strange nonchaotic attractors described by Grebogi et al and Herman in the mid-1980s have some chaotic properties. More precisely, we show that these systems exhibit sensitive dependence on initial conditions, both on the whole phase space and restricted to the attractor. The results also remain valid in more general classes of quasiperiodically forced systems. Further, we include an elementary proof of a classic result by Glasner and Weiss on sensitive dependence, and we clarify the structure of the attractor in an example with two-dimensional fibers also introduced by Grebogi et al.
Chaotic attractors of two-dimensional invertible maps
Discrete Dynamics in Nature and Society, 1998
In this paper, we investigate the characteristics of quasihyperbolic attractors and quasiattractors in invertible dissipative maps of the plane. The criteria which allow one to diagnose the indicated types of attractors in numerical experiments are formulated.
Influence of noise on statistical properties of nonhyperbolic attractors
Physical Review E, 2000
We analyze effects of bounded white and colored noise on nonhyperbolic chaotic attractors in twodimensional invertible maps. It is shown that first the nonhyperbolic nature is kept even in the presence of strong noise, but secondly already due to weak noise some properties of nonhyperbolic chaos can become similar to those of hyperbolic and almost hyperbolic chaos. We also estimate the stationary probability measure of noisy nonhyperbolic attractors. For this purpose two different methods for calculating the probability density are applied and the obtained results are compared in detail.