The Lorenz Attractor is Mixing

By Stefano Luzzatto, Ian Melbourne, Frédéric Paccaut. 2004

133 citations recorded by the source corpus.

We study a class of geometric Lorenz flows, introduced independently by Afraimovič, Bykov & Sil′nikov and by Guckenheimer & Williams, and give a verifiable condition for such flows to be mixing. As a consequence, we show that the classical Lorenz attractor is mixing.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/cee0922c35ca1a868cbb50b62bf96f364bc0fd14

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In Control and Systems Engineering

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Homoclinic and heteroclinic bifurcations in vector fields (2010, unknown): “We refer to [6, 12, 15, 262] for further results, addressing primarily ergodic properties, Lorenz-like and other singular hyperbolic attractors.”
Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator (2016, background): “This attractor is proven to have the mixing property which implies ergodicity [23].”
Singular-hyperbolic attractors are chaotic (2005, background): “In all these cases the results show that these attractors are expansive and have physical measures which are u-Gibbs states.”
Data-driven spectral analysis of the Koopman operator (2017, result): “This is in agreement with the fact that the Lorenz system is mixing (see [24]) and hence there are no non-trivial eigenvalues of the Koopman operator.”
Central limit theorems and invariance principles for Lorenz attractors (2007, background): “Oddly, the logarithmic singularity of h is is a crucial part of the proof of mixing in [9], yet is the main obstruction to proving statistical limit laws or rapid decay of correlations.”
Lorenz-like flows: exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence (2009, methodology): “We remark that the speed of correlation decay for Lorenz like flows is not yet known (although some are proved to be mixing, see [29]) hence quantitative recurrence and hitting time results cannot be proved directly using this tool, instead of this we will consider a Poincaré section, estimating its”
What Are the New Implications of Chaos for Unpredictability? (2009, background): “It has only been proven that there is chaotic behaviour near hyperbolic fixed points, where the motion is indeed mixing (Moser [1973], chapter 3).”