Statistics of the stochastically-forced Lorenz attractor by the Fokker-Planck equation and cumulant expansions

TL;DR

This paper explores the statistical properties of the Lorenz-63 attractor under stochastic forcing using Fokker-Planck equations and cumulant expansions, providing new computational methods and comparisons with numerical simulations.

Contribution

It introduces a novel approach to compute the invariant measure of the stochastic Lorenz system via sparse linear algebra and cumulant expansions, enhancing understanding of its statistical behavior.

Findings

Invariant measure computed via Fokker-Planck operator

Comparison shows cumulant expansion approximates numerical results

Hyperdiffusion variant improves computational stability

Abstract

We investigate the Fokker-Planck description of the equal-time statistics of the three-dimensional Lorenz-63 attractor with additive white noise. The invariant measure is found by computing the zero (or null) mode of the linear Fokker-Planck operator as a problem of sparse linear algebra. Two variants are studied: A self-adjoint construction of the linear operator, and the replacement of diffusion with hyperdiffusion. We also access the low-order statistics of the system by a perturbative expansion in equal-time cumulants. Comparison is made to statistics obtained by the standard approach of accumulation via direct numerical simulation. Theoretical and computational aspects of the Fokker-Planck and cumulant expansion methods are discussed.