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

We investigate the Fokker-Planck description of the equal-time statistics of the three-dimensional Lorenz 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. A 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.

Figure 1

(a) PDF of the unforced Lorenz system (binned on a ${637}^3$ grid) accumulated by DNS. Classic Lorenz parameters are used (see Sec. 2). (b) Same as (a), but with added stochastic forcing $(\mathrm{\Gamma }=0.2)$ $({478}^3$ grid). Note that the fine rings visible in the unforced system are washed out by the noise. (c) PDF of the stochastically forced attractor $\left(\mathrm{\Gamma }=0.2\right)$ as obtained from the zero mode of ${\widehat{L}}_{\text{FPE}}$ on a ${160}^3$ grid and for $x\in [-12.5,12.5],y\in [-24,24]$, and $z\in [1,45]$. The JDQR algorithm is employed. Rows (1), (2), and (3) correspond to the $x\text{−}y,x\text{−}z$, and $y\text{−}z$ projections of the PDF, respectively. Good agreement between (b) DNS and (c) FPE is evident. Red lines correspond to the cross sections shown in Fig. 2.

Figure 2

(a) Vertical and (b) horizontal slices along the red lines of the $x\text{−}y$ planar projection shown in Fig. 1. The DNS and FPE methods agree quantitatively.

Figure 3

(a) PDF of the unforced Lorenz system (binned on a ${637}^3$ grid) accumulated by DNS. Modified parameters are used (see Sec. 2). (b) Same as (a), but with added stochastic forcing $(\mathrm{\Gamma }=0.02)$ $({478}^3$ grid). Note that the fine rings visible in the unforced system are washed out by the noise. (c) PDF of the stochastically forced attractor $\left(\mathrm{\Gamma }=0.02\right)$ as obtained from the zero mode of ${\widehat{L}}_{\text{FPE}}$ on a ${160}^3$ grid and for $x\in [-7,7],y\in [-10,10]$, and $z\in [15,35]$. Rows (1), (2), and (3) correspond to the $x\text{−}y,x\text{−}z$, and $y\text{−}z$ projections of the PDF, respectively. Good agreement between (b) DNS and (c) FPE is evident. Red lines correspond to the cross sections shown in Fig. 4 and green lines correspond to the cross section shown in Fig. 5.

Figure 4

(a) Vertical and (b) horizontal slices along the red lines of the $y\text{−}z$ planar projection shown in Fig. 3. The DNS and FPE methods agree quantitatively.

Figure 5

Cross section through the $x\text{−}y$ planar projection of the PDF where $x=-2.77$ for unforced DNS, stochastically forced DNS, and a stochastically forced FPE (with $\mathrm{\Gamma }=0.02)$. Modified parameters are used. Sharp peaks in probability seen in the unforced Lorenz system are eliminated by the stochastic forcing.

Figure 6

Real component of the first 12 nonzero eigenvalues of the linear FPE operator of the Lorenz system (on an ${80}^3$ lattice grid) as a function of stochastic forcing $\mathrm{\Gamma }$. Modified parameters are used (see Sec. 2) with $x\in [-12.5,12.5],y\in [-17.5,17.5]$, and $z\in [7.5,42.5]$. The phase of each eigenvalue is represented by the color of the points, with white corresponding to a phase of zero. A linear extrapolation of the first nonzero eigenvalue to $\mathrm{\Gamma }=0$ is shown. The second and third nonzero eigenvalues are purely real.