.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fractals_chaos/maps/plot_01_henon_attractor.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_fractals_chaos_maps_plot_01_henon_attractor.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_fractals_chaos_maps_plot_01_henon_attractor.py:


The Hénon attractor
=========================

Iterates Hénon's map (x, y) -> (1 - 1.4 x^2 + y, 0.3 x), zooms in on
its strange attractor to reveal repeated layers, and estimates its
fractal and Lyapunov characteristics.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.fractals_chaos import box_counting_dimension, henon_map








.. GENERATED FROM PYTHON SOURCE LINES 17-19

The attractor and a zoom
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-29

.. code-block:: Python


    orbit = henon_map(300000)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    axes[0].plot(*orbit.T, ",k", alpha=0.5)
    axes[0].add_patch(plt.Rectangle((0.55, 0.15), 0.1, 0.05, fill=False, color="tab:red"))
    axes[0].set_title("Hénon attractor")
    zoom = orbit[(orbit[:, 0] > 0.55) & (orbit[:, 0] < 0.65) & (orbit[:, 1] > 0.15) & (orbit[:, 1] < 0.2)]
    axes[1].plot(*zoom.T, ",k")
    axes[1].set_title("zoom: the lines split into bundles of lines")




.. image-sg:: /api/gallery/fractals_chaos/maps/images/sphx_glr_plot_01_henon_attractor_001.png
   :alt: Hénon attractor, zoom: the lines split into bundles of lines
   :srcset: /api/gallery/fractals_chaos/maps/images/sphx_glr_plot_01_henon_attractor_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 1.0, 'zoom: the lines split into bundles of lines')



.. GENERATED FROM PYTHON SOURCE LINES 30-32

Dimension and sensitivity
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-47

.. code-block:: Python


    print(f"box-counting estimate {box_counting_dimension(orbit).dimension:.3f}; careful studies give about 1.26, and coarse box sizes run high")

    a, b = 1.4, 0.3
    x, y = 0.1, 0.1
    v = np.array([1.0, 0.0])
    total = 0.0
    steps = 100000
    for _ in range(steps):
        v = np.array([[-2 * a * x, 1.0], [b, 0.0]]) @ v
        x, y = 1 - a * x * x + y, b * x
        norm = np.linalg.norm(v)
        total += np.log(norm)
        v /= norm
    print(f"largest Lyapunov exponent ~ {total / steps:.3f} (positive: chaotic)")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    box-counting estimate 1.338; careful studies give about 1.26, and coarse box sizes run high
    largest Lyapunov exponent ~ 0.421 (positive: chaotic)





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 1.516 seconds)


.. _sphx_glr_download_api_gallery_fractals_chaos_maps_plot_01_henon_attractor.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_henon_attractor.ipynb <plot_01_henon_attractor.ipynb>`

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

      :download:`Download Python source code: plot_01_henon_attractor.py <plot_01_henon_attractor.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_henon_attractor.zip <plot_01_henon_attractor.zip>`


.. only:: html

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

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_