Lorenz 96 model

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

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

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

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

Python simulation

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]
  # add the forcing term
  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

↑ Lorenz, Edward (1996). "Predictability – A problem partly solved" (PDF). Seminar on Predictability, Vol. I, ECMWF.
↑ Ott, Edward. "A Local Ensemble Kalman Filter for Atmospheric Data Assimilation".

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