# 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.