# Lorenz 96 model

The Lorenz 96 model is a dynamical system formulated by Edward Lorenz in 1996. 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 $x_{N+1}=x_{1}$ and $N\geq 4$ . 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.

## Python simulation

from mpl_toolkits.mplot3d import Axes3D
from scipy.integrate import odeint
import matplotlib.pyplot as plt
import numpy as np

# These are our constants
N = 5  # Number of variables
F = 8  # Forcing

def L96(x, t):
"""Lorenz 96 model with constant forcing"""
# Setting up vector
d = np.zeros(N)
# Loops over indices (with operations and Python underflow indexing handling edge cases)
for i in range(N):
d[i] = (x[(i + 1) % N] - x[i - 2]) * x[i - 1] - x[i] + F
return d

x0 = F * np.ones(N)  # Initial state (equilibrium)
x0 += 0.01  # Add small perturbation to the first variable
t = np.arange(0.0, 30.0, 0.01)

x = odeint(L96, x0, t)

# Plot the first three variables
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()


## Julia simulation

using DynamicalSystems, PyPlot
PyPlot.using3D()

# parameters and initial conditions
N = 5
F = 8.0
u₀ = F * ones(N)
u₀ += 0.01 # small perturbation

# The Lorenz-96 model is predefined in DynamicalSystems.jl:
ds = Systems.lorenz96(N; F = F)

# Equivalently, to define a fast version explicitly, do:
struct Lorenz96{N} end # Structure for size type
function (obj::Lorenz96{N})(dx, x, p, t) where {N}
F = p
# 3 edge cases explicitly (performance)
@inbounds dx = (x - x[N - 1]) * x[N] - x + F
@inbounds dx = (x - x[N]) * x - x + F
@inbounds dx[N] = (x - x[N - 2]) * x[N - 1] - x[N] + F
# then the general case
for n in 3:(N - 1)
@inbounds dx[n] = (x[n + 1] - x[n - 2]) * x[n - 1] - x[n] + F
end
return nothing
end

lor96 = Lorenz96{N}() # create struct
ds = ContinuousDynamicalSystem(lor96, u₀, [F])

# And now evolve a trajectory
dt = 0.01 # sampling time
Tf = 30.0 # final time
tr = trajectory(ds, Tf; dt = dt)

# And plot in 3D:
x, y, z = columns(tr)
plot3D(x, y, z)