Direct statistical simulation of the Lorenz63 system.

We use direct statistical simulation to find the low-order statistics of the well-known dynamical system, the Lorenz63 model. Instead of accumulating statistics from numerical simulation of the dynamical system or solving the Fokker-Planck equation for the full probability distribution of the dynamical system, we directly solve the equations of motion for the low-order statistics after closing them by making several different choices for the truncation. Fixed points of the statistics are obtained either by time evolving or by iterative methods. The stability and statistical realizability of the fixed points of the statistics are analyzed, and the statistics so obtained are compared to those found by the traditional approach. Low-order statistics of the chaotic Lorenz63 system can be obtained from cumulant expansions more efficiently than by accumulation via direct numerical simulation or by solution of the Fokker-Planck equation.