.. 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_02_bendixson_criterion.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_02_bendixson_criterion.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_ode_dynamics_limit_cycles_plot_02_bendixson_criterion.py:


Bendixson's negative criterion
====================================

If the divergence of a planar vector field keeps one sign on a simply
connected region, no closed orbit fits inside it. A damped oscillator
passes the test everywhere; the Van der Pol field fails it because its
divergence :math:`\mu(1 - x^2)` changes sign at :math:`|x| = 1` -- and
indeed its limit cycle must cross those lines.

.. GENERATED FROM PYTHON SOURCE LINES 13-22

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.ode_dynamics.systems.limit_cycles import VanDerPolOscillator, bendixson_criterion

    systems = {
        "damped oscillator": lambda x, y: (y, -x - 0.4 * y),
        "Van der Pol (mu=1)": lambda x, y: (y, (1.0 - x**2) * y - x),
    }








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Divergence maps
---------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-42

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    for ax, (name, f) in zip(axes, systems.items()):
        res = bendixson_criterion(f, (-3, 3), (-3, 3), n=201)
        lim = max(abs(res.divergence.min()), abs(res.divergence.max()))
        mesh = ax.pcolormesh(res.X, res.Y, res.divergence, cmap="RdBu_r", vmin=-lim, vmax=lim, shading="auto")
        fig.colorbar(mesh, ax=ax, label="divergence")
        ax.set_title(f"{name}: cycles ruled out = {res.rules_out_periodic_orbits}")
        print(f"{name}: rules out periodic orbits -> {res.rules_out_periodic_orbits}")

    traj = VanDerPolOscillator([0.5, 0.0], mu=1.0).integrate((0.0, 40.0), dt=1e-3, method="rk4")
    axes[1].plot(traj.y[:, 0], traj.y[:, 1], "k", lw=1)
    axes[1].set_xlim(-3, 3)
    axes[1].set_ylim(-3, 3)
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/ode_dynamics/limit_cycles/images/sphx_glr_plot_02_bendixson_criterion_001.png
   :alt: damped oscillator: cycles ruled out = True, Van der Pol (mu=1): cycles ruled out = False
   :srcset: /api/gallery/ode_dynamics/limit_cycles/images/sphx_glr_plot_02_bendixson_criterion_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    damped oscillator: rules out periodic orbits -> True
    Van der Pol (mu=1): rules out periodic orbits -> False





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

   **Total running time of the script:** (0 minutes 0.103 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_limit_cycles_plot_02_bendixson_criterion.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_bendixson_criterion.ipynb <plot_02_bendixson_criterion.ipynb>`

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

      :download:`Download Python source code: plot_02_bendixson_criterion.py <plot_02_bendixson_criterion.py>`

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

      :download:`Download zipped: plot_02_bendixson_criterion.zip <plot_02_bendixson_criterion.zip>`


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

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