.. 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_duffing_oscillator.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_duffing_oscillator.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_chaos_continuous_systems_plot_duffing_oscillator.py:


Duffing Oscillator
==================

The forced, damped Duffing oscillator models a particle in a double-well
potential driven by a periodic force,

.. math::

    \ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos(\omega t),

or, as the first-order system in :math:`(x, v)` actually integrated,
:math:`\dot{x} = v`, :math:`\dot{v} = -\delta v - \alpha x - \beta x^3 +
\gamma \cos(\omega t)`. Here :math:`\delta` is the damping coefficient,
:math:`\alpha` and :math:`\beta` set the (linear and cubic) restoring force
-- with :math:`\alpha<0` and :math:`\beta>0`, as used below, the potential
:math:`V(x)=\tfrac12\alpha x^2+\tfrac14\beta x^4` has two wells -- and
:math:`\gamma`, :math:`\omega` are the forcing amplitude and frequency. For
the classic chaotic parameter set below, its long-time trajectory settles
onto a strange attractor in the ``(x, v)`` phase plane.

.. GENERATED FROM PYTHON SOURCE LINES 22-30

.. code-block:: Python


    import matplotlib.pyplot as plt

    from physicskit.chaos.systems.continuous import Duffing
    from physicskit.chaos.utils.dimension import box_counting_dimension, correlation_dimension

    system = Duffing(delta=0.3, alpha=-1.0, beta=1.0, gamma=0.37, omega=1.2)








.. GENERATED FROM PYTHON SOURCE LINES 31-35

Integrate
---------
Discard an initial transient so the plotted trajectory has already settled
onto the attractor.

.. GENERATED FROM PYTHON SOURCE LINES 35-38

.. code-block:: Python

    t, states = system.trajectory(n_steps=40000, dt=0.02)
    states = states[5000:]








.. GENERATED FROM PYTHON SOURCE LINES 39-41

Phase portrait
---------------

.. GENERATED FROM PYTHON SOURCE LINES 41-49

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(7, 6))
    ax.plot(states[:, 0], states[:, 1], lw=0.2, color="crimson")
    ax.set_xlabel("x")
    ax.set_ylabel("v")
    ax.set_title("Duffing oscillator: chaotic attractor")

    plt.show()




.. image-sg:: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_duffing_oscillator_001.png
   :alt: Duffing oscillator: chaotic attractor
   :srcset: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_duffing_oscillator_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 50-60

Quantifying it: a fractal attractor
----------------------------------------
The dense-but-structured phase portrait above is a strange attractor: its
box-counting (capacity) dimension :math:`D_0` and correlation dimension
:math:`D_2` (see :doc:`/api/gallery/chaos/chaos_metrics/plot_fractal_dimension`
for both estimators applied to the Henon map) are non-integers, between the
curve (:math:`D=1`) a single trajectory traces out locally and the
2D plane it never quite fills. Only every 4th stroboscopic-ish sample is
kept for the correlation-dimension estimate, since it computes all
:math:`O(n^2)` pairwise distances.

.. GENERATED FROM PYTHON SOURCE LINES 60-78

.. code-block:: Python

    d0, eps0, counts0 = box_counting_dimension(states)
    d2, eps2, csum2 = correlation_dimension(states[::4])
    print(f"Duffing attractor: box-counting D0 = {d0:.3f}, correlation D2 = {d2:.3f}")

    fig2, axes2 = plt.subplots(1, 2, figsize=(12, 5))
    axes2[0].loglog(1.0 / eps0, counts0, "o-", color="steelblue")
    axes2[0].set_xlabel(r"$1/\epsilon$")
    axes2[0].set_ylabel(r"$N(\epsilon)$")
    axes2[0].set_title(f"Box counting: D0 = {d0:.3f}")

    axes2[1].loglog(eps2, csum2, "o-", color="darkorange")
    axes2[1].set_xlabel(r"$\epsilon$")
    axes2[1].set_ylabel(r"$C(\epsilon)$")
    axes2[1].set_title(f"Correlation sum: D2 = {d2:.3f}")
    fig2.suptitle("Duffing attractor: fractal dimension between 1 (curve) and 2 (plane)")
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_duffing_oscillator_002.png
   :alt: Duffing attractor: fractal dimension between 1 (curve) and 2 (plane), Box counting: D0 = 1.180, Correlation sum: D2 = 1.301
   :srcset: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_duffing_oscillator_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Duffing attractor: box-counting D0 = 1.180, correlation D2 = 1.301





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

   **Total running time of the script:** (0 minutes 1.064 seconds)


.. _sphx_glr_download_api_gallery_chaos_continuous_systems_plot_duffing_oscillator.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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