"""

================

Lorenz attractor

================

This is an example of plotting Edward Lorenz's 1963 `"Deterministic Nonperiodic

Flow"`_ in a 3-dimensional space using mplot3d.

.. _"Deterministic Nonperiodic Flow":

https://journals.ametsoc.org/view/journals/atsc/20/2/1520-0469_1963_020_0130_dnf_2_0_co_2.xml

.. note::

Because this is a simple non-linear ODE, it would be more easily done using

SciPy's ODE solver, but this approach depends only upon NumPy.

"""

import matplotlib.pyplot as plt

import numpy as np

def lorenz(xyz, *, s=10, r=28, b=2.667):

"""

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 np.array([x_dot, y_dot, z_dot])

dt = 0.01

num_steps = 10000

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):

xyzs[i + 1] = xyzs[i] + lorenz(xyzs[i]) * dt

# Plot

ax = plt.figure().add_subplot(projection='3d')

ax.plot(*xyzs.T, lw=0.5)

ax.set_xlabel("X Axis")

ax.set_ylabel("Y Axis")

ax.set_zlabel("Z Axis")

ax.set_title("Lorenz Attractor")

plt.show()

# %%

# .. tags::

# plot-type: 3D,

# level: intermediate