

.. _sphx_glr_gallery_mplot3d_lorenz_attractor.py:


================
Lorenz Attractor
================

This is an example of plotting Edward Lorenz's 1963 `"Deterministic
Nonperiodic Flow"
<http://journals.ametsoc.org/doi/abs/10.1175/1520-0469%281963%29020%3C0130%3ADNF%3E2.0.CO%3B2>`_
in a 3-dimensional space using mplot3d.

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.




.. image:: /gallery/mplot3d/images/sphx_glr_lorenz_attractor_001.png
    :align: center





.. code-block:: python


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


    def lorenz(x, y, z, 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
        '''
        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


    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)

    # 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)


    # Plot
    fig = plt.figure()
    ax = fig.gca(projection='3d')

    ax.plot(xs, ys, zs, 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()

**Total running time of the script:** ( 0 minutes  0.127 seconds)



.. container:: sphx-glr-footer


  .. container:: sphx-glr-download

     :download:`Download Python source code: lorenz_attractor.py <lorenz_attractor.py>`



  .. container:: sphx-glr-download

     :download:`Download Jupyter notebook: lorenz_attractor.ipynb <lorenz_attractor.ipynb>`

.. rst-class:: sphx-glr-signature

    `Generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_
