.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/ode_dynamics/limit_cycles/plot_01_van_der_pol.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_limit_cycles_plot_01_van_der_pol.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_ode_dynamics_limit_cycles_plot_01_van_der_pol.py:


The Van der Pol limit cycle
================================

Trajectories starting both inside and outside the eventual limit cycle
converge onto the *same* closed orbit -- the defining property that
distinguishes a limit cycle from a linear center (whose orbit amplitude
is set entirely by the initial condition).

.. GENERATED FROM PYTHON SOURCE LINES 12-17

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.ode_dynamics.systems.limit_cycles import VanDerPolOscillator, estimate_limit_cycle_amplitude
    from mathematicskit.ode_dynamics.visualizers.plots import plot_phase_portrait








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Trajectories from very different initial conditions converge
-----------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-30

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(6, 6))
    for state0 in ([0.1, 0.0], [4.0, 0.0], [0.0, 3.0]):
        system = VanDerPolOscillator(state0, mu=1.0)
        traj = system.integrate((0.0, 30.0), dt=1e-3, method="rk4")
        plot_phase_portrait(traj, ax=ax, lw=0.8, label=f"start={state0}")
    ax.legend()
    ax.set_title("Van der Pol (mu=1): all orbits approach the same limit cycle")
    fig.tight_layout()




.. image-sg:: /api/gallery/ode_dynamics/limit_cycles/images/sphx_glr_plot_01_van_der_pol_001.png
   :alt: Van der Pol (mu=1): all orbits approach the same limit cycle
   :srcset: /api/gallery/ode_dynamics/limit_cycles/images/sphx_glr_plot_01_van_der_pol_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 31-33

Amplitude is independent of mu's transient details
--------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-39

.. code-block:: Python


    for mu in (0.5, 1.0, 2.0, 5.0):
        amp = estimate_limit_cycle_amplitude(mu=mu, t_transient=150.0, t_observe=40.0)
        print(f"mu={mu}: limit-cycle amplitude ~ {amp:.4f}")

    plt.show()




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

 .. code-block:: none

    mu=0.5: limit-cycle amplitude ~ 2.0025
    mu=1.0: limit-cycle amplitude ~ 2.0086
    mu=2.0: limit-cycle amplitude ~ 2.0199
    mu=5.0: limit-cycle amplitude ~ 2.0215





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

   **Total running time of the script:** (0 minutes 1.067 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_limit_cycles_plot_01_van_der_pol.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_van_der_pol.ipynb <plot_01_van_der_pol.ipynb>`

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

      :download:`Download Python source code: plot_01_van_der_pol.py <plot_01_van_der_pol.py>`

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

      :download:`Download zipped: plot_01_van_der_pol.zip <plot_01_van_der_pol.zip>`


.. only:: html

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

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