.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        Click :ref:`here <sphx_glr_download_gallery_mplot3d_lorenz_attractor.py>`     to download the full example code
    .. rst-class:: sphx-glr-example-title

    .. _sphx_glr_gallery_mplot3d_lorenz_attractor.py:


================
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":
   http://journals.ametsoc.org/doi/abs/10.1175/1520-0469%281963%29020%3C0130%3ADNF%3E2.0.CO%3B2

.. 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
    :alt: Lorenz Attractor
    :class: sphx-glr-single-img






.. code-block:: default


    import numpy as np
    import matplotlib.pyplot as plt


    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()


.. _sphx_glr_download_gallery_mplot3d_lorenz_attractor.py:


.. only :: html

 .. container:: sphx-glr-footer
    :class: sphx-glr-footer-example



  .. container:: sphx-glr-download sphx-glr-download-python

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



  .. container:: sphx-glr-download sphx-glr-download-jupyter

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


.. only:: html

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

    Keywords: matplotlib code example, codex, python plot, pyplot
    `Gallery generated by Sphinx-Gallery
    <https://sphinx-gallery.readthedocs.io>`_