.. 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_driven_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_driven_pendulum.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_chaos_continuous_systems_plot_driven_pendulum.py:


Driven Pendulum: the Period-Doubling Route to Chaos
========================================================

The sinusoidally forced, damped pendulum obeys

.. math::

    \ddot\theta + \gamma \dot\theta + \frac{g}{l}\sin\theta
        = A \cos(\omega_d t),

or, as the first-order system in :math:`(\theta, \omega)` actually
integrated, :math:`\dot\theta = \omega`, :math:`\dot\omega = -\gamma\omega -
(g/l)\sin\theta + A\cos(\omega_d t)`, where :math:`\gamma` is the viscous
damping coefficient, :math:`g/l` is the small-angle frequency squared of the
undamped, undriven pendulum, and :math:`A`, :math:`\omega_d` are the forcing
torque's amplitude and angular frequency. It is the classic mechanical
system for the period-doubling route to chaos: D'Humieres, Beasley,
Huberman, and Libchaber (1982) showed experimentally that sweeping the
forcing amplitude :math:`A` reproduces the same cascade of period-doubling
bifurcations as the logistic map, this time in a genuine forced mechanical
oscillator. At the default (deep-chaotic) parameters below, the pendulum's
swing never settles into a repeating pattern.

.. GENERATED FROM PYTHON SOURCE LINES 25-34

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.chaos.systems.continuous import DrivenPendulum
    from physicskit.chaos.visualizers.bifurcation import plot_bifurcation_diagram, stroboscopic_bifurcation_sampler

    system = DrivenPendulum(damping=0.5, g_over_l=1.0, A=1.5, omega_d=2.0 / 3.0)








.. GENERATED FROM PYTHON SOURCE LINES 35-37

A single trajectory
-----------------------

.. GENERATED FROM PYTHON SOURCE LINES 37-47

.. code-block:: Python

    t, states = system.trajectory(n_steps=4000, dt=0.02)
    theta = states[:, 0]

    fig, ax = plt.subplots(figsize=(8, 4))
    ax.plot(t, theta, lw=0.6, color="indigo")
    ax.set_xlabel("t")
    ax.set_ylabel(r"$\theta(t)$")
    ax.set_title("Driven pendulum: no repeating pattern at these parameters")
    fig.tight_layout()




.. image-sg:: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_driven_pendulum_001.png
   :alt: Driven pendulum: no repeating pattern at these parameters
   :srcset: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_driven_pendulum_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 48-58

The period-doubling route to chaos
----------------------------------------
Sampling the angular velocity omega once per forcing period turns the
continuous flow into an effective discrete map -- the same
stroboscopic-sampling trick used for the Duffing oscillator (omega, unlike
theta, stays bounded even while the pendulum occasionally rotates over the
top). Sweeping the forcing amplitude A over this narrow window reveals the
same period-doubling cascade Feigenbaum found in the abstract logistic
map -- period-1, then period-2, period-4, and a chaotic band, interrupted
by a periodic window -- now in a physically forced pendulum.

.. GENERATED FROM PYTHON SOURCE LINES 58-79

.. code-block:: Python

    period = 2.0 * np.pi / system.omega_d

    sampler = stroboscopic_bifurcation_sampler(
        lambda A: DrivenPendulum(damping=0.5, g_over_l=1.0, A=A, omega_d=2.0 / 3.0),
        state0=np.array([0.2, 0.0]),
        sample_period=period,
        n_transient_periods=60,
        n_keep_periods=16,
        component=1,
        dt=0.1,
        rtol=1e-6,
        atol=1e-8,
    )

    A_values = np.linspace(1.30, 1.50, 100)
    fig2, ax2 = plot_bifurcation_diagram(A_values, sampler, n_jobs=4, marker=".", markersize=1.0)
    ax2.set_xlabel(r"forcing amplitude $A$")
    ax2.set_ylabel(r"$\omega$ (stroboscopic samples)")
    ax2.set_title("Driven pendulum: period-doubling route to chaos")

    plt.show()



.. image-sg:: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_driven_pendulum_002.png
   :alt: Driven pendulum: period-doubling route to chaos
   :srcset: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_driven_pendulum_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 25.857 seconds)


.. _sphx_glr_download_api_gallery_chaos_continuous_systems_plot_driven_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_driven_pendulum.ipynb <plot_driven_pendulum.ipynb>`

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

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

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

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


.. only:: html

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

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