.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fractals_chaos/mandelbrot_julia/plot_01_julia_sets.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_mandelbrot_julia_plot_01_julia_sets.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_fractals_chaos_mandelbrot_julia_plot_01_julia_sets.py:


Fatou and Julia: Julia sets of z -> z^2 + c
=================================================

Iterates the quadratic map :math:`z \mapsto z^2 + c` from every starting
point :math:`z_0` of a grid. Points whose orbits stay bounded form the
filled Julia set; its boundary, the Julia set, separates the Fatou set,
where nearby orbits behave alike, from the region where orbits escape
to infinity. Changing :math:`c` turns the Julia set from a circle into
a fractal curve or a disconnected dust.

.. GENERATED FROM PYTHON SOURCE LINES 14-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.fractals_chaos import julia_set
    from mathematicskit.fractals_chaos.visualizers.plots import plot_escape_time








.. GENERATED FROM PYTHON SOURCE LINES 21-28

Four values of c
----------------

For :math:`c = 0` the Julia set is the unit circle. The other three are
fractal: the "basilica" (:math:`c=-1`) and Douady's "rabbit" are
connected, while for :math:`c` outside the Mandelbrot set the Julia set
breaks up into a totally disconnected dust.

.. GENERATED FROM PYTHON SOURCE LINES 28-44

.. code-block:: Python


    params = {
        "c = 0 (circle)": 0.0 + 0.0j,
        "c = -1 (basilica)": -1.0 + 0.0j,
        "c = -0.123 + 0.745i (rabbit)": -0.123 + 0.745j,
        "c = 0.4 + 0.4i (dust)": 0.4 + 0.4j,
    }
    fig, axes = plt.subplots(2, 2, figsize=(8, 8))
    for ax, (label, c) in zip(axes.ravel(), params.items()):
        result = julia_set(c=c, extent=(-1.6, 1.6, -1.6, 1.6), resolution=300, max_iter=200)
        plot_escape_time(result, ax=ax, cmap="magma")
        ax.set_title(label)
        bounded = np.mean(result.iterations == result.max_iter)
        print(f"{label:30s} fraction of grid with bounded orbits: {bounded:.3f}")
    fig.tight_layout()




.. image-sg:: /api/gallery/fractals_chaos/mandelbrot_julia/images/sphx_glr_plot_01_julia_sets_001.png
   :alt: c = 0 (circle), c = -1 (basilica), c = -0.123 + 0.745i (rabbit), c = 0.4 + 0.4i (dust)
   :srcset: /api/gallery/fractals_chaos/mandelbrot_julia/images/sphx_glr_plot_01_julia_sets_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    c = 0 (circle)                 fraction of grid with bounded orbits: 0.305
    c = -1 (basilica)              fraction of grid with bounded orbits: 0.137
    c = -0.123 + 0.745i (rabbit)   fraction of grid with bounded orbits: 0.127
    c = 0.4 + 0.4i (dust)          fraction of grid with bounded orbits: 0.000




.. GENERATED FROM PYTHON SOURCE LINES 45-52

Sensitive dependence near the Julia set
---------------------------------------

Two starting points a distance :math:`10^{-6}` apart, straddling the
unit circle (the Julia set for :math:`c = 0`), end up in different
places: one orbit tends to 0, the other escapes. Two points in the same
Fatou component stay together.

.. GENERATED FROM PYTHON SOURCE LINES 52-60

.. code-block:: Python


    for z0 in (1.0 - 5e-7, 1.0 + 5e-7, 0.5, 0.5 + 1e-6):
        z = complex(z0)
        for _ in range(40):
            z = z * z
            if abs(z) > 1e6:
                break
        print(f"z0 = {z0:.7f}: |z_n| -> {'infinity' if abs(z) > 1e6 else f'{abs(z):.1e}'}")




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

 .. code-block:: none

    z0 = 0.9999995: |z_n| -> 0.0e+00
    z0 = 1.0000005: |z_n| -> infinity
    z0 = 0.5000000: |z_n| -> 0.0e+00
    z0 = 0.5000010: |z_n| -> 0.0e+00





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

   **Total running time of the script:** (0 minutes 0.216 seconds)


.. _sphx_glr_download_api_gallery_fractals_chaos_mandelbrot_julia_plot_01_julia_sets.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_julia_sets.ipynb <plot_01_julia_sets.ipynb>`

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

      :download:`Download Python source code: plot_01_julia_sets.py <plot_01_julia_sets.py>`

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

      :download:`Download zipped: plot_01_julia_sets.zip <plot_01_julia_sets.zip>`


.. only:: html

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

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