.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/ode_dynamics/poincare/plot_01_duffing_poincare_section.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_ode_dynamics_poincare_plot_01_duffing_poincare_section.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_ode_dynamics_poincare_plot_01_duffing_poincare_section.py:


Poincare sections of the driven Duffing oscillator
========================================================

Sampling the Duffing oscillator's state once per drive period
(stroboscopically) collapses its effectively-3D state space (position,
velocity, and drive phase) back to a 2D picture: a periodic response
shows up as one (or a few) fixed points, while chaotic parameters
produce a scattered, fractal-looking set of points.

.. GENERATED FROM PYTHON SOURCE LINES 13-18

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.ode_dynamics.systems.poincare import DuffingOscillator, stroboscopic_poincare_section
    from mathematicskit.ode_dynamics.visualizers.plots import plot_poincare_points








.. GENERATED FROM PYTHON SOURCE LINES 19-21

A weakly driven system settles onto a single periodic point
------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-25

.. code-block:: Python


    system_periodic = DuffingOscillator([0.1, 0.0], delta=0.5, alpha=1.0, beta=0.0, gamma=0.05, omega=1.0)
    xs_p, ys_p = stroboscopic_poincare_section(system_periodic, n_periods=40, n_transient_periods=40, dt=1e-2)








.. GENERATED FROM PYTHON SOURCE LINES 26-28

A more strongly driven double-well system shows a richer section
-----------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-40

.. code-block:: Python


    system_rich = DuffingOscillator([1.0, 0.0], delta=0.3, alpha=-1.0, beta=1.0, gamma=0.37, omega=1.2)
    xs_r, ys_r = stroboscopic_poincare_section(system_rich, n_periods=400, n_transient_periods=200, dt=1e-2)

    fig, axes = plt.subplots(1, 2, figsize=(10, 5))
    plot_poincare_points(xs_p, ys_p, ax=axes[0])
    axes[0].set_title("Weakly driven: converges to a point")
    plot_poincare_points(xs_r, ys_r, ax=axes[1])
    axes[1].set_title("Strongly driven: richer structure")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/ode_dynamics/poincare/images/sphx_glr_plot_01_duffing_poincare_section_001.png
   :alt: Weakly driven: converges to a point, Strongly driven: richer structure
   :srcset: /api/gallery/ode_dynamics/poincare/images/sphx_glr_plot_01_duffing_poincare_section_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.605 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_poincare_plot_01_duffing_poincare_section.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_duffing_poincare_section.ipynb <plot_01_duffing_poincare_section.ipynb>`

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

      :download:`Download Python source code: plot_01_duffing_poincare_section.py <plot_01_duffing_poincare_section.py>`

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

      :download:`Download zipped: plot_01_duffing_poincare_section.zip <plot_01_duffing_poincare_section.zip>`


.. only:: html

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

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