diff --git a/examples/animation/double_pendulum.py b/examples/animation/double_pendulum.py
index 8f50f6cb195d..fba7ea4ec9e2 100644
--- a/examples/animation/double_pendulum.py
+++ b/examples/animation/double_pendulum.py
@@ -12,7 +12,6 @@
from numpy import sin, cos
import numpy as np
import matplotlib.pyplot as plt
-import scipy.integrate as integrate
import matplotlib.animation as animation
from collections import deque
@@ -22,13 +21,13 @@
L = L1 + L2 # maximal length of the combined pendulum
M1 = 1.0 # mass of pendulum 1 in kg
M2 = 1.0 # mass of pendulum 2 in kg
-t_stop = 5 # how many seconds to simulate
+t_stop = 2.5 # how many seconds to simulate
history_len = 500 # how many trajectory points to display
-def derivs(state, t):
-
+def derivs(t, state):
dydx = np.zeros_like(state)
+
dydx[0] = state[1]
delta = state[2] - state[0]
@@ -51,7 +50,7 @@ def derivs(state, t):
return dydx
# create a time array from 0..t_stop sampled at 0.02 second steps
-dt = 0.02
+dt = 0.01
t = np.arange(0, t_stop, dt)
# th1 and th2 are the initial angles (degrees)
@@ -64,8 +63,15 @@ def derivs(state, t):
# initial state
state = np.radians([th1, w1, th2, w2])
-# integrate your ODE using scipy.integrate.
-y = integrate.odeint(derivs, state, t)
+# integrate the ODE using Euler's method
+y = np.empty((len(t), 4))
+y[0] = state
+for i in range(1, len(t)):
+ y[i] = y[i - 1] + derivs(t[i - 1], y[i - 1]) * dt
+
+# A more accurate estimate could be obtained e.g. using scipy:
+#
+# y = scipy.integrate.solve_ivp(derivs, t[[0, -1]], state, t_eval=t).y.T
x1 = L1*sin(y[:, 0])
y1 = -L1*cos(y[:, 0])
diff --git a/examples/mplot3d/lorenz_attractor.py b/examples/mplot3d/lorenz_attractor.py
index 921bf5a47099..7162c12d2dce 100644
--- a/examples/mplot3d/lorenz_attractor.py
+++ b/examples/mplot3d/lorenz_attractor.py
@@ -18,45 +18,41 @@
import matplotlib.pyplot as plt
-def lorenz(x, y, z, s=10, r=28, b=2.667):
+def lorenz(xyz, *, s=10, r=28, b=2.667):
"""
- Given:
- x, y, z: a point of interest in three dimensional space
- s, r, b: parameters defining the lorenz attractor
- Returns:
- x_dot, y_dot, z_dot: values of the lorenz attractor's partial
- derivatives at the point x, y, z
+ Parameters
+ ----------
+ xyz : array-like, shape (3,)
+ Point of interest in three dimensional space.
+ s, r, b : float
+ Parameters defining the Lorenz attractor.
+
+ Returns
+ -------
+ xyz_dot : array, shape (3,)
+ Values of the Lorenz attractor's partial derivatives at *xyz*.
"""
+ x, y, z = xyz
x_dot = s*(y - x)
y_dot = r*x - y - x*z
z_dot = x*y - b*z
- return x_dot, y_dot, z_dot
+ return np.array([x_dot, y_dot, z_dot])
dt = 0.01
num_steps = 10000
-# Need one more for the initial values
-xs = np.empty(num_steps + 1)
-ys = np.empty(num_steps + 1)
-zs = np.empty(num_steps + 1)
-
-# Set initial values
-xs[0], ys[0], zs[0] = (0., 1., 1.05)
-
+xyzs = np.empty((num_steps + 1, 3)) # Need one more for the initial values
+xyzs[0] = (0., 1., 1.05) # Set initial values
# Step through "time", calculating the partial derivatives at the current point
# and using them to estimate the next point
for i in range(num_steps):
- x_dot, y_dot, z_dot = lorenz(xs[i], ys[i], zs[i])
- xs[i + 1] = xs[i] + (x_dot * dt)
- ys[i + 1] = ys[i] + (y_dot * dt)
- zs[i + 1] = zs[i] + (z_dot * dt)
-
+ xyzs[i + 1] = xyzs[i] + lorenz(xyzs[i]) * dt
# Plot
ax = plt.figure().add_subplot(projection='3d')
-ax.plot(xs, ys, zs, lw=0.5)
+ax.plot(*xyzs.T, lw=0.5)
ax.set_xlabel("X Axis")
ax.set_ylabel("Y Axis")
ax.set_zlabel("Z Axis")
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