  Simultaneous ensemble mean equations (LEMEs) for the Lorenz model are
obtained, enabling us to analyze the properties of the ensemble mean from a
dynamical point of view. The qualitative analysis for the two-sample and
n-sample LEMEs show the locations and number of stable points are different
from the Lorenz equations (LEs), and the results are validated by numerical
experiments. The analysis for the eigenmatrix of the stable points of LEMEs
indicates that the stability of these stable points is similar to the LEs'. The
eigenmatrix for non-stable points can be obtained too, but the eigenvalues
depend not only on the value of the mean variable but also the other n-1 sample
equation's variable, and thus for these points there may be different
stabilities compared to the LEs'. The divergence of the LEMEs' flow has a
negative value, which is the same as the LEs', and thus the trajectory in phase
space approaches zero and the trajectory will be attracted to a low-level
dimensional curved surface, i.e., the LEMEs have the attractor property, but
the structure of the attractor is not the same as the LEs'.
