Lorenz 96 model

The Lorenz 96 model is a dynamical system formulated by Edward Lorenz in 1996.[1] It is defined as follows. For ${\displaystyle i=1,...,N}$:

${\displaystyle {\frac {dx_{i}}{dt}}=(x_{i+1}-x_{i-2})x_{i-1}-x_{i}+F}$

where it is assumed that ${\displaystyle x_{-1}=x_{N-1},x_{0}=x_{N}}$ and ${\displaystyle x_{N+1}=x_{1}}$. Here ${\displaystyle x_{i}}$ is the state of the system and ${\displaystyle F}$ is a forcing constant. ${\displaystyle F=8}$ is a common value known to cause chaotic behavior.

It is commonly used as a model problem in data assimilation.[2]

Python simulation

Plot of the first three variables of the simulation
from scipy.integrate import odeint
import matplotlib.pyplot as plt
import numpy as np

# these are our constants
N = 36  # number of variables
F = 8  # forcing

def Lorenz96(x,t):

# compute state derivatives
d = np.zeros(N)
# first the 3 edge cases: i=1,2,N
d[0] = (x[1] - x[N-2]) * x[N-1] - x[0]
d[1] = (x[2] - x[N-1]) * x[0]- x[1]
d[N-1] = (x[0] - x[N-3]) * x[N-2] - x[N-1]
# then the general case
for i in range(2, N-1):
d[i] = (x[i+1] - x[i-2]) * x[i-1] - x[i]
d = d + F

# return the state derivatives
return d

x0 = F*np.ones(N) # initial state (equilibrium)
x0[19] += 0.01 # add small perturbation to 20th variable
t = np.arange(0.0, 30.0, 0.01)

x = odeint(Lorenz96, x0, t)

# plot first three variables
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot(x[:,0],x[:,1],x[:,2])
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
ax.set_zlabel('$x_3$')
plt.show()


References

1. ^ Lorenz, Edward (1996). "Predictability – A problem partly solved" (PDF). Seminar on Predictability, Vol. I, ECMWF.
2. ^ Ott, Edward; et al. "A Local Ensemble Kalman Filter for Atmospheric Data Assimilation". arXiv:physics/0203058.