Rapid Mixing for the Lorenz Attractor and Statistical Limit Laws for Their Time-1 Maps

2013/11/30 by V. Araujo, V. Araújo, I. Melbourne +1
Engineering · Mathematics · Physics and Astronomy · #Attractor #Central limit theorem #Chaos control and synchronization #Flow (mathematics) #Invariance principle #Limit (mathematics) #Lorenz system #Mathematical Dynamics and Fractals #Mixing (physics) #Rössler attractor #Stability and Controllability of Differential Equations #math-ph #math.CA #math.DS #math.MP #msc:37A25 #msc:37A50 #msc:37D30 #msc:37D45

paper · pdf · doi:10.1007/s00220-015-2471-0

published as Commun. Math. Phys. 340, 901-938 (2015) · 30 pages, 3 figures. Final accepted version to appear in Communications in Math Phys

arxiv created 2015/09/04 · openalex publication_date 2015/09/22 · openalex created_date 2016/06/24 · arxiv updated 2016/11/24 · openalex updated_date 2026/08/05

Abstract

We prove that every geometric Lorenz attractor has superpolynomial decay of correlations with respect to the unique SRB measure. Moreover, we prove the Central Limit Theorem and Almost Sure Invariance Principle for the time-1 map of the flow of geometric Lorenz attractors. In particular, our results apply to the classical Lorenz attractor.

Citations

• Almost sure invariance principles for partial sums of weakly dependent random variables
• An Invariance Principle for Maps with Polynomial Decay of Correlations
• Statistical Properties of Dynamical Systems with Some Hyperbolicity
• Structural stability of Lorenz attractors
• What’s new on lorenz strange attractors?
• Singular-hyperbolic attractors are chaotic
• A Rigorous ODE Solver and Smale’s 14th Problem
• Deterministic Nonperiodic Flow