Friends,
I thought you'd like to see the solution.
Many thanks to Jake Vanderplas for his code and teachings:
https://jakevdp.github.io/blog/2013/02/16/animating-the-lorentz-system-in-3d/
If you start a new IP Notebook session, run as your first entry:
%pylab
and then copy and paste the text below and run it, you should be good to go
(on a Mac, at least).
There are several parameters I've changed from his original, and I've
commented as I've changed. The original code is at the link above.
There is one error in his code -- I've documented it below.
Again, thanks to the community, Jake, and Ben Root.
--Prahas
···
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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
# orig value of N_traj was 20 -- very cool this way.
N_trajectories = 1
def lorentz_deriv((x, y, z), t0, sigma=10., beta=8./3, rho=28.0):
"""Compute the time-derivative of a Lorentz system."""
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)
# changing from -15,30 to 10,5 below starts the drawing in the middle,
# rather than getting the long line from below....
# if using N_Traj > 1, return to orig values.
# x0 = -15 + 30 * np.random.random((N_trajectories, 3))
x0 = 10 + 5 * np.random.random((N_trajectories, 3))
# Solve for the trajectories
# orig values: 0,4,1000
# 3rd value -- lower it, it gets choppier.
# 2nd value -- increase it -- more points, but speedier.
# change middle num from 4 to 15 -- this adds points!!!!!!!!
t = np.linspace(0, 40, 3000)
x_t = np.asarray([integrate.odeint(lorentz_deriv, x0i, t)
for x0i in x0])
# Set up figure & 3D axis for animation
fig = plt.figure()
ax = fig.add_axes([0, 0, 1, 1], projection='3d')
# changing off to on below adds axises. slows it down but you
# can fix that with interval value in the animation call
ax.axis('on')
# choose a different color for each trajectory
colors = plt.cm.jet(np.linspace(0, 1, N_trajectories))
# set up lines and points -- this is a correction from
# the orig jake code. the next four lines...
lines = [ax.plot([], [], [], '-', c=c)[0]
for c in colors]
pts = [ax.plot([], [], [], 'o', c=c)[0]
for c in colors]
# prepare the axes limits
ax.set_xlim((-25, 25))
ax.set_ylim((-35, 35))
ax.set_zlim((5, 55))
# set point-of-view: specified by (altitude degrees, azimuth degrees)
ax.view_init(30, 0)
# initialization function: plot the background of each frame
def init():
for line, pt in zip(lines, pts):
line.set_data([], [])
line.set_3d_properties([])
pt.set_data([], [])
pt.set_3d_properties([])
return lines + pts
# animation function. This will be called sequentially with the frame number
def animate(i):
# we'll step two time-steps per frame. This leads to nice results.
i = (2 * i) % x_t.shape[1]
for line, pt, xi in zip(lines, pts, x_t):
x, y, z = xi[:i].T
line.set_data(x, y)
line.set_3d_properties(z)
pt.set_data(x[-1:], y[-1:])
pt.set_3d_properties(z[-1:])
# changed 0.3 to 0.05 below -- this slows the rotation of the view.
# changed 30 to 20 below
# changing 20 to (20 + (.1 * i)) rotates on the Z axis. trippy.
ax.view_init(10, 0.1 * i)
# ax.view_init(10, 100)
fig.canvas.draw()
return lines + pts
# instantiate the animator. I've deleted the blit switch (for Mac)
# enlarging frames=500 works now -- it failed before because I didn't give it
# enough data -- by changing the t=np.linspace line above I generate
more points.
# interval larger slows it down
# changed inteval from 30 to 200, frames from 500 to 3000
anim = animation.FuncAnimation(fig, animate, init_func=init,
frames=3000, interval=200)
# Save as mp4. This requires mplayer or ffmpeg to be installed. COMPLEX!!!!!
# Instead, use a screen record program: Quicktime on the Mac; MS
Expression Encoder on PC.
# anim.save('PDNlorentz_attractor.mp4', fps=15, extra_args=['-vcodec',
'libx264'])
plt.show()
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