In this Notebook we explore the Lorenz system of differential equations:
This is one of the classic systems in non-linear differential equations. It exhibits a range of different behaviors as the parameters (\(\sigma\), \(\beta\), \(\rho\)) are varied.
First, we import the needed things from IPython, NumPy, Matplotlib and SciPy.
In [1]:
%matplotlib inline
In [2]:
from ipywidgets import interact, interactive
from IPython.display import clear_output, display, HTML
In [3]:
import numpy as np
from scipy import integrate
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.colors import cnames
from matplotlib import animation
We define a function that can integrate the differential equations
numerically and then plot the solutions. This function has arguments
that control the parameters of the differential equation
(\(\sigma\), \(\beta\), \(\rho\)), the numerical integration
(N
, max_time
) and the visualization (angle
).
In [4]:
def solve_lorenz(N=10, angle=0.0, max_time=4.0, sigma=10.0, beta=8./3, rho=28.0):
fig = plt.figure()
ax = fig.add_axes([0, 0, 1, 1], projection='3d')
ax.axis('off')
# prepare the axes limits
ax.set_xlim((-25, 25))
ax.set_ylim((-35, 35))
ax.set_zlim((5, 55))
def lorenz_deriv(x_y_z, t0, sigma=sigma, beta=beta, rho=rho):
"""Compute the time-derivative of a Lorenz system."""
x, y, z = x_y_z
return [sigma * (y - x), x * (rho - z) - y, x * y - beta * z]
# Choose random starting points, uniformly distributed from -15 to 15
np.random.seed(1)
x0 = -15 + 30 * np.random.random((N, 3))
# Solve for the trajectories
t = np.linspace(0, max_time, int(250*max_time))
x_t = np.asarray([integrate.odeint(lorenz_deriv, x0i, t)
for x0i in x0])
# choose a different color for each trajectory
colors = plt.cm.jet(np.linspace(0, 1, N))
for i in range(N):
x, y, z = x_t[i,:,:].T
lines = ax.plot(x, y, z, '-', c=colors[i])
plt.setp(lines, linewidth=2)
ax.view_init(30, angle)
plt.show()
return t, x_t
Let’s call the function once to view the solutions. For this set of parameters, we see the trajectories swirling around two points, called attractors.
In [5]:
t, x_t = solve_lorenz(angle=0, N=10)
Using IPython’s interactive
function, we can explore how the
trajectories behave as we change the various parameters.
In [6]:
w = interactive(solve_lorenz, angle=(0.,360.), N=(0,50), sigma=(0.0,50.0), rho=(0.0,50.0))
display(w)
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1.88188188, 1.88588589, 1.88988989, 1.89389389, 1.8978979 ,
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1.92192192, 1.92592593, 1.92992993, 1.93393393, 1.93793794,
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1.98198198, 1.98598599, 1.98998999, 1.99399399, 1.997998 ,
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3.56356356, 3.56756757, 3.57157157, 3.57557558, 3.57957958,
3.58358358, 3.58758759, 3.59159159, 3.5955956 , 3.5995996 ,
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3.68368368, 3.68768769, 3.69169169, 3.6956957 , 3.6996997 ,
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3.76376376, 3.76776777, 3.77177177, 3.77577578, 3.77977978,
3.78378378, 3.78778779, 3.79179179, 3.7957958 , 3.7997998 ,
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3.82382382, 3.82782783, 3.83183183, 3.83583584, 3.83983984,
3.84384384, 3.84784785, 3.85185185, 3.85585586, 3.85985986,
3.86386386, 3.86786787, 3.87187187, 3.87587588, 3.87987988,
3.88388388, 3.88788789, 3.89189189, 3.8958959 , 3.8998999 ,
3.9039039 , 3.90790791, 3.91191191, 3.91591592, 3.91991992,
3.92392392, 3.92792793, 3.93193193, 3.93593594, 3.93993994,
3.94394394, 3.94794795, 3.95195195, 3.95595596, 3.95995996,
3.96396396, 3.96796797, 3.97197197, 3.97597598, 3.97997998,
3.98398398, 3.98798799, 3.99199199, 3.995996 , 4. ]),
array([[[ -2.48933986e+00, 6.60973480e+00, -1.49965688e+01],
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[ 9.03371359e+00, 1.00685518e+01, 2.62975123e+01]],
...,
[[ 1.40478473e+01, -5.59727466e+00, 5.76967847e+00],
[ 1.33015844e+01, -4.35137818e+00, 5.43762841e+00],
[ 1.26324566e+01, -3.15868945e+00, 5.18607122e+00],
...,
[ -5.62183444e+00, -9.08227752e+00, 1.57576353e+01],
[ -5.76241216e+00, -9.32397964e+00, 1.57989149e+01],
[ -5.90705639e+00, -9.57059699e+00, 1.58506576e+01]],
[[ 1.12916746e+01, 1.18381999e+01, -1.24486737e+01],
[ 1.13481065e+01, 1.36039497e+01, -1.17430178e+01],
[ 1.14710812e+01, 1.53431579e+01, -1.09606825e+01],
...,
[ -1.47677316e-02, -1.23193351e-02, 1.01977111e+01],
[ -1.46914304e-02, -1.33208729e-02, 1.00894066e+01],
[ -1.46573919e-02, -1.43208005e-02, 9.98225246e+00]],
[[ -1.38283565e+01, -9.90508741e+00, 1.13442751e+01],
[ -1.36915606e+01, -1.07690154e+01, 1.17903119e+01],
[ -1.35933180e+01, -1.15964765e+01, 1.22727276e+01],
...,
[ 1.61306741e+01, 1.93584734e+01, 3.43088188e+01],
[ 1.62473985e+01, 1.88446346e+01, 3.51764615e+01],
[ 1.63382575e+01, 1.82747191e+01, 3.60076113e+01]]]))
The object returned by interactive
is a Widget
object and it has
attributes that contain the current result and arguments:
In [7]:
t, x_t = w.result
In [8]:
w.kwargs
Out[8]:
{'N': 10,
'angle': 0.0,
'beta': 2.6666666666666665,
'max_time': 4.0,
'rho': 28.0,
'sigma': 10.0}
After interacting with the system, we can take the result and perform further computations. In this case, we compute the average positions in \(x\), \(y\) and \(z\).
In [9]:
xyz_avg = x_t.mean(axis=1)
In [10]:
xyz_avg.shape
Out[10]:
(10, 3)
Creating histograms of the average positions (across different trajectories) show that on average the trajectories swirl about the attractors.
In [11]:
plt.hist(xyz_avg[:,0])
plt.title('Average $x(t)$')
Out[11]:
<matplotlib.text.Text at 0x7f54f8033e48>
In [12]:
plt.hist(xyz_avg[:,1])
plt.title('Average $y(t)$')
Out[12]:
<matplotlib.text.Text at 0x7f54f83a3518>