.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/chaos/maps/plot_henon_map.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_chaos_maps_plot_henon_map.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_chaos_maps_plot_henon_map.py:


Henon Map
=========

The Henon map is a simple 2D quadratic map,

.. math::

    x_{n+1} = 1 - a x_n^2 + y_n, \qquad y_{n+1} = b x_n

whose classic parameters (:math:`a=1.4`, :math:`b=0.3`) produce a strange
attractor with the map's signature fractal, self-similar structure. Unlike
the (area-preserving) standard and baker's maps, the Henon map is
dissipative -- its Jacobian determinant is the constant :math:`-b`, with
:math:`|b|<1` -- so nearby points contract in area on average even while
they stretch apart along the attractor's unstable direction. This example
iterates a single long trajectory and scatter-plots it.

.. GENERATED FROM PYTHON SOURCE LINES 19-29

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.chaos.systems.maps import HenonMap
    from physicskit.chaos.visualizers import theme
    from physicskit.chaos.visualizers.bifurcation import map_bifurcation_sampler, plot_bifurcation_diagram

    system = HenonMap(a=1.4, b=0.3)








.. GENERATED FROM PYTHON SOURCE LINES 30-36

The finished attractor
--------------------------
Discard a short initial transient so the plotted points already lie on the
attractor, then plot the full, much longer orbit at once to reveal the
map's signature fractal, self-similar structure -- especially visible when
zooming into the folds.

.. GENERATED FROM PYTHON SOURCE LINES 36-47

.. code-block:: Python

    traj = system.trajectory(np.array([0.0, 0.0]), n_iter=50000)
    traj = traj[100:]

    fig, ax = plt.subplots(figsize=(7, 5))
    ax.scatter(traj[:, 0], traj[:, 1], s=0.3, color=theme.PRIMARY, alpha=0.6)
    ax.set_xlabel("x")
    ax.set_ylabel("y")
    ax.set_title("Henon map attractor")

    plt.show()




.. image-sg:: /api/gallery/chaos/maps/images/sphx_glr_plot_henon_map_001.png
   :alt: Henon map attractor
   :srcset: /api/gallery/chaos/maps/images/sphx_glr_plot_henon_map_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 48-59

The route there: a bifurcation diagram over a
---------------------------------------------------
The single attractor above is only the end state at one fixed ``a``
(``b`` held at its classic 0.3). Sweeping ``a`` and, for each value,
plotting the surviving long-term ``x`` values -- exactly the same
:func:`~physicskit.chaos.visualizers.bifurcation.plot_bifurcation_diagram`
machinery used for the :doc:`Logistic Map </api/gallery/chaos/maps/plot_logistic_map>`
-- shows *how* that attractor is reached: a period-doubling cascade out of
simple fixed points and cycles, opening into the fractal chaotic band at
the classic ``a = 1.4`` used above (dashed line), interrupted by visible
periodic windows.

.. GENERATED FROM PYTHON SOURCE LINES 59-70

.. code-block:: Python

    a_values = np.linspace(0.8, 1.42, 1200)
    sampler = map_bifurcation_sampler(lambda a: HenonMap(a=a, b=0.3), state0=np.array([0.0, 0.0]), n_transient=300, n_keep=150)

    fig2, ax2 = plot_bifurcation_diagram(a_values, sampler)
    ax2.axvline(1.4, color=theme.MUTED, ls="--", lw=1.0, label="a = 1.4 (shown above)")
    ax2.set_xlabel("a")
    ax2.set_ylabel("x (long-term values)")
    ax2.set_title("Henon map bifurcation diagram (b = 0.3)")
    ax2.legend(fontsize=8)

    plt.show()



.. image-sg:: /api/gallery/chaos/maps/images/sphx_glr_plot_henon_map_002.png
   :alt: Henon map bifurcation diagram (b = 0.3)
   :srcset: /api/gallery/chaos/maps/images/sphx_glr_plot_henon_map_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.082 seconds)


.. _sphx_glr_download_api_gallery_chaos_maps_plot_henon_map.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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