Random Perturbations of SRB Measures and Numerical Studies of Chaotic Dynamics

The goal of this thesis was to explore whether noise can reduce correlation times in chaotic dynamical systems, with the hope of improving the convergence rates of numerically-computed time averages. The short answer is that noise can help reduce initialization bias but does not reduce integrated correlation time.

Here is the original abstract:

Chaotic behavior occurs naturally in a variety of physical situations governed by deterministic equations of motion. Deterministic chaos is characterized by the exponential separation of nearby initial conditions. Thus chaotic systems are intrinsically unstable, and this intrinsic instability makes the equations of motion computationally intractable over long times. In contrast, the frequencies with which a solution visits different states is generally stable under small perturbations of the solution and of the equations of motion, so the corresponding statistical averages can be extracted from long time numerical simulations.

This thesis proposes a number of simple algorithms which compute statistical averages of observables by adding noise to deterministic chaotic systems. The results of this study show that the addition of noise can be beneficial in numerical studies of the statistics of chaotic systems: in addition to covering up numerical artifacts which arise from round-off errors, a moderate amount of noise can help accelerate the convergence of computed time averages of observable quantities. A simple scaling argument is used to derive a rough error estimate and the unique ergodicity of the perturbed process is proved. The effect of noise on the statistical properties of the Kuramoto-Sivashinsky equation, in particular the Lyapunov exponents and the statistical dependence of Fourier modes, is also studied numerically.

Download. You can download the entire thesis in PDF (130 pages).

This page was last updated on June 07, 2012.