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

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

.. _sphx_glr_api_gallery_chaos_continuous_systems_plot_magnetic_pendulum.py:


Magnetic Pendulum: Fractal Basins of Attraction
====================================================

The magnetic pendulum is a damped bob, modeled in the small-swing (flat, 2D)
limit at position :math:`(x, y)`, pulled toward the origin by a linear
restoring force and toward each of :math:`N` fixed magnets at
:math:`(m_{x,k}, m_{y,k})` by a softened inverse-square force:

.. math::

    \ddot{x} &= -\gamma \dot{x} - \kappa x - \sum_{k=1}^{N} S
        \frac{x - m_{x,k}}{r_k^3} \\
    \ddot{y} &= -\gamma \dot{y} - \kappa y - \sum_{k=1}^{N} S
        \frac{y - m_{y,k}}{r_k^3} \\
    r_k &= \sqrt{(x - m_{x,k})^2 + (y - m_{y,k})^2 + h^2}

where :math:`\gamma` is friction, :math:`\kappa` the restoring-force
("spring") coefficient, :math:`S` the magnet strength, and :math:`h` the
bob's height above the magnet plane (which softens the pull so it never
truly diverges directly above a magnet). This is the classic system for
visualizing *fractal basin boundaries*: which magnet the bob eventually
settles near depends on its starting position with famously fractal
sensitivity. A tiny nudge in initial position can flip the outcome
entirely, even though the bob's long-term motion is not chaotic itself (it
always settles down) -- the *sensitivity to initial conditions* lives
entirely in the geometry of the basin boundaries.

.. GENERATED FROM PYTHON SOURCE LINES 29-39

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.chaos.systems.continuous import MagneticPendulum, magnetic_pendulum_rhs
    from physicskit.chaos.visualizers.basins import plot_basin_of_attraction

    system = MagneticPendulum()
    state0 = np.array([0.5, 0.5, 0.0, 0.0])








.. GENERATED FROM PYTHON SOURCE LINES 40-43

A single trajectory
-----------------------
Released off-center, the bob spirals in and settles near one magnet.

.. GENERATED FROM PYTHON SOURCE LINES 43-54

.. code-block:: Python

    t, states = system.trajectory(state0=state0, n_steps=5000, dt=0.02)

    fig, ax = plt.subplots(figsize=(6, 6))
    ax.plot(states[:, 0], states[:, 1], lw=0.6, color="steelblue")
    ax.plot(*system.magnet_positions.T, "o", color="black", markersize=10)
    ax.plot(states[0, 0], states[0, 1], "s", color="crimson", label="start")
    ax.plot(states[-1, 0], states[-1, 1], "*", color="gold", markersize=15, label="settled")
    ax.set_aspect("equal")
    ax.legend()
    ax.set_title("Magnetic pendulum: a single trajectory")




.. image-sg:: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_magnetic_pendulum_001.png
   :alt: Magnetic pendulum: a single trajectory
   :srcset: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_magnetic_pendulum_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Magnetic pendulum: a single trajectory')



.. GENERATED FROM PYTHON SOURCE LINES 55-61

The basin of attraction map
--------------------------------
For every point on a grid of starting positions, integrate to find out
which magnet the bob settles near, and color each pixel accordingly. The
resulting boundary between colors is a fractal -- zooming into any edge
reveals more fine structure, forever.

.. GENERATED FROM PYTHON SOURCE LINES 61-72

.. code-block:: Python

    fig2, ax2 = plot_basin_of_attraction(
        magnetic_pendulum_rhs,
        system.params,
        x_range=(-2.0, 2.0),
        y_range=(-2.0, 2.0),
        attractors=system.magnet_positions,
        resolution=300,
    )
    ax2.set_title("Magnetic pendulum: basin of attraction")

    plt.show()



.. image-sg:: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_magnetic_pendulum_002.png
   :alt: Magnetic pendulum: basin of attraction
   :srcset: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_magnetic_pendulum_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 9.460 seconds)


.. _sphx_glr_download_api_gallery_chaos_continuous_systems_plot_magnetic_pendulum.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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