Journal ArticleDOI
Unstable periodic orbits and the dimensions of multifractal chaotic attractors
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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 is pursued.Abstract:
The probability measure generated by typical chaotic orbits of a dynamical system can have an arbitrarily fine-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 repellers are considered.read more
Citations
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The analysis of observed chaotic data in physical systems
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Estimating fractal dimension
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Controlling Cardiac Chaos
TL;DR: By administering electrical stimuli to the heart at irregular times determined by chaos theory, the arrhythmia was converted to periodic beating and was stabilized to stabilize cardiac arrhythmias induced by the drug ouabain in rabbit ventricle.
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The control of chaos: theory and applications
TL;DR: In this paper, the Ott-Grebogi-Yorke (OGY) method and the adaptive method for chaotic control are discussed. But the authors focus on the targeting problem, i.e., how to bring a trajectory to a small neighborhood of a desired location in the chaotic attractor in both low and high dimensions.
References
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Fractal measures and their singularities: The characterization of strange sets
TL;DR: A description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory, focusing upon the scaling properties of such measures, which are characterized by two indices: \ensuremath{\alpha}, which determines the strength of their singularities; and f, which describes how densely they are distributed.
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Fractal measures and their singularities: The characterization of strange sets
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The infinite number of generalized dimensions of fractals and strange attractors
TL;DR: In this article, it was shown that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions Dq, q > 0, which correspond to exponents associated with ternary, quaternary and higher correlation functions.
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Crises, sudden changes in chaotic attractors, and transient chaos
TL;DR: In this article, the authors show that crisis events are prevalent in many circumstances and systems, and that, just past a crisis, certain characteristic statistical behavior (whose type depends on the type of crisis) occurs.
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The dimension of chaotic attractors
TL;DR: In this paper, the authors discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors, and conclude that dimension of the natural measure is more important than the fractal dimension.