.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/kinetics/oscillators/plot_01_brusselator.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_kinetics_oscillators_plot_01_brusselator.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_kinetics_oscillators_plot_01_brusselator.py:


Prigogine and Lefever's Brusselator limit cycle
==================================================

Below the Hopf-bifurcation threshold (:math:`B < 1 + A^2`) the
Brusselator's single fixed point is a stable focus and any initial
concentration relaxes toward it. Above threshold, that same fixed point
becomes unstable and every trajectory is drawn instead onto a surrounding
stable limit cycle -- sustained chemical oscillation with no external
forcing, driven purely by the autocatalytic :math:`X^2 Y` term.

.. GENERATED FROM PYTHON SOURCE LINES 14-21

.. code-block:: Python

    import matplotlib.pyplot as plt

    from chemistrykit.kinetics.systems.oscillators import Brusselator
    from chemistrykit.kinetics.visualizers.kinetics_plots import plot_concentration_vs_time, plot_phase_portrait

    fig, axes = plt.subplots(2, 2, figsize=(11, 8))




.. image-sg:: /api/gallery/kinetics/oscillators/images/sphx_glr_plot_01_brusselator_001.png
   :alt: plot 01 brusselator
   :srcset: /api/gallery/kinetics/oscillators/images/sphx_glr_plot_01_brusselator_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 22-23

Below threshold: relaxation to the fixed point (X*, Y*) = (A, B/A).

.. GENERATED FROM PYTHON SOURCE LINES 23-32

.. code-block:: Python

    below = Brusselator(X0=2.0, Y0=0.2, A=1.0, B=1.5)
    result_below = below.integrate((0.0, 60.0), dt=1e-2, method="rk4")
    plot_concentration_vs_time(result_below, ax=axes[0, 0])
    axes[0, 0].set_title(f"Below threshold (B={below.B} < 1+A^2={1 + below.A**2}): relaxes to fixed point")
    plot_phase_portrait(result_below, "X", "Y", ax=axes[0, 1])
    fp_below = below.fixed_point()
    axes[0, 1].plot(*fp_below, "k*", markersize=12, label="fixed point")
    axes[0, 1].legend()





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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11950fe00>



.. GENERATED FROM PYTHON SOURCE LINES 33-34

Above threshold: a stable limit cycle around the now-unstable fixed point.

.. GENERATED FROM PYTHON SOURCE LINES 34-45

.. code-block:: Python

    above = Brusselator(X0=1.0, Y0=1.0, A=1.0, B=3.0)
    result_above = above.integrate((0.0, 60.0), dt=1e-2, method="rk4")
    plot_concentration_vs_time(result_above, ax=axes[1, 0])
    axes[1, 0].set_title(f"Above threshold (B={above.B} > 1+A^2={1 + above.A**2}): sustained oscillation")
    plot_phase_portrait(result_above, "X", "Y", ax=axes[1, 1])
    fp_above = above.fixed_point()
    axes[1, 1].plot(*fp_above, "k*", markersize=12, label="unstable fixed point")
    axes[1, 1].legend()

    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.304 seconds)


.. _sphx_glr_download_api_gallery_kinetics_oscillators_plot_01_brusselator.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_brusselator.ipynb <plot_01_brusselator.ipynb>`

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

      :download:`Download Python source code: plot_01_brusselator.py <plot_01_brusselator.py>`

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

      :download:`Download zipped: plot_01_brusselator.zip <plot_01_brusselator.zip>`


.. only:: html

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

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