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

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.

Learning Force Fields from Stochastic Trajectories

“…• Fourier modes of order n = 0...7; specifically, we use all functions of the form cos 2π µ k µ (x µ − x µ )/R µ and sin 2π µ k µ (x µ − x µ )/R µ with non-negative integers k µ such that µ k µ ≤ n. Here we choose R µ to be 1.05 times the diameter of the trajectory in direction µ. Our second nonlinear process is a stochastic variant of a popular model for dynamical systems, the Lorenz system [26]. Its 3D Brownian dynamics is described by the force field…”

Adjoint sensitivity analysis of chaotic systems using cumulant truncation

“…If the functional J, and therefore the vector ∂ U (1) J ≡ g (1) , is specified then one can solve for the adjoint variables v (1) according to the second row of (24). However, a consistency requirement for the extended system (24) to possess a solution is that g (2) = T (12) † v (1) 0, in general. We are therefore not at liberty to choose the functional J arbitrarily, because it will automatically contain a contribution scaled by g (2) from the unclosed perturbations u (2) i .…”

“…However, a consistency requirement for the extended system (24) to possess a solution is that g (2) = T (12) † v (1) 0, in general. We are therefore not at liberty to choose the functional J arbitrarily, because it will automatically contain a contribution scaled by g (2) from the unclosed perturbations u (2) i . The vacuous consequence of using (24) is that only functionals whose value can be determined identically from the original cumulant equations, such as equation (15), can be determined exactly.…”

“…Statistical state dynamics, or direct statistical simulation [46,1] has therefore been applied with success in simulations of planetary jets [37,47,14,7] and wall-bounded shear-flow [17,13]. Whilst strongly nonlinear systems, such as the model for Rayleigh-Bénard convection given by the Lorenz equations [31], require a more sophisticated treatment that accounts for the role of cumulants beyond second order, direct statistical simulation can nevertheless produce accurate predictions [2].…”

A data-driven method for the steady state of randomly perturbed dynamics

“…Take the Lorenz system for an example. A very expensive numerical computation in [2] can only solve the Fokker-Planck equation corresponding to the Lorenz system on a 160×160×160 mesh, with a grid size ≈ 0.3.…”