2002/09/18 by Albert Fannjiang, Fannjiang, Albert, Lech Wołowski +2 · 2 citations
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS
paper · pdf · doi:10.48550/arxiv.math/0209231
The research is supported in part by the grant from U.S. National Science Foundation, DMS-9971322 and Lech Wolowski
openalex publication_date 2002/09/18 · arxiv created 2003/05/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider dissipative systems resulting from the Gaussian and alpha-stable noise perturbations of measure-preserving maps on the d dimensional torus. We study the dissipation time scale and its physical implications as the noise level \vep vanishes. We show that nonergodic maps give rise to an O(1/\vep) dissipation time whereas ergodic toral automorphisms, including cat maps and their d-dimensional generalizations, have an O(ln(1/\vep)) dissipation time with a constant related to the minimal, \em dimensionally averaged entropy among the automorphism's irreducible blocks. Our approach reduces the calculation of the dissipation time to a nonlinear, arithmetic optimization problem which is solved asymptotically by means of some fundamental theorems in theories of convexity, Diophantine approximation and arithmetic progression. We show that the same asymptotic can be reproduced by degenerate noises as well as mere coarse-graining. We also discuss the implication of the dissipation time in kinematic dynamo.