Journal ArticleDOI
Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions
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The authors obtained similar results for equivariant observations on compact group extensions of hyperbolic basic sets using the Ruelle transfer operator, and showed that such observations satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles.Abstract:
Holder continuous observations on hyperbolic basic sets satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles (approximation by Brownian motion). Using an equivariant version of the Ruelle transfer operator studied by Parry and Pollicott, we obtain similar results for equivariant observations on compact group extensions of hyperbolic basic sets.read more
Citations
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A new test for chaos in deterministic systems
Georg A. Gottwald,Ian Melbourne +1 more
TL;DR: In this paper, a new test for determining whether a given deterministic dynamical system is chaotic or non-chaotic is proposed, which is applied directly to the time-series data and does not require phase space reconstruction.
Journal ArticleDOI
A New Test for Chaos
Georg A. Gottwald,Ian Melbourne +1 more
TL;DR: In this article, a new test for determining whether a given deterministic dynamical system is chaotic or non-chaotic is proposed, which is independent of the dimension of the system.
Journal ArticleDOI
On the Implementation of the 0-1 Test for Chaos
Georg A. Gottwald,Ian Melbourne +1 more
TL;DR: Practical aspects of the implementation of the 0–1 test for chaos in deterministic systems are addressed and a new formulation of the test is presented which significantly increases its sensitivity.
Journal ArticleDOI
On the Implementation of the 0-1 Test for Chaos
Georg A. Gottwald,Ian Melbourne +1 more
TL;DR: In this article, the authors present a new formulation of the 0-1 test for chaos in deterministic systems, which significantly increases its sensitivity and can be viewed as a method to distill a binary quantity from the power spectrum.
Journal ArticleDOI
Almost Sure Invariance Principle for Nonuniformly Hyperbolic Systems
Ian Melbourne,Matthew Nicol +1 more
TL;DR: In this paper, the authors prove an almost sure invariance principle for general classes of nonuniformly expanding and non-uneiformly hyperbolic dynamical systems, and apply it to planar periodic Lorentz flows with finite horizon.
References
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Journal ArticleDOI
An Introduction to Probability Theory and Its Applications
David A. Freedman,William Feller +1 more
Book
Convergence of Probability Measures
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
An Introduction To Probability Theory And Its Applications
TL;DR: A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.
Book
Probability and Measure
TL;DR: In this paper, the convergence of distributions is considered in the context of conditional probability, i.e., random variables and expected values, and the probability of a given distribution converging to a certain value.