.. 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_03_brusselator_hopf.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_03_brusselator_hopf.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_ode_dynamics_limit_cycles_plot_03_brusselator_hopf.py:


The Brusselator's Hopf bifurcation
========================================

The Brusselator's fixed point :math:`(a, b/a)` is stable until :math:`b`
crosses :math:`b_c = 1 + a^2`, where the trace of its Jacobian changes
sign and a limit cycle is born: the model chemical reaction starts to
oscillate.

.. GENERATED FROM PYTHON SOURCE LINES 12-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.ode_dynamics.systems.chemical_oscillators import Brusselator, brusselator_hopf_threshold

    a = 1.0
    b_c = brusselator_hopf_threshold(a)
    print(f"Hopf threshold b_c = 1 + a^2 = {b_c}")





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

 .. code-block:: none

    Hopf threshold b_c = 1 + a^2 = 2.0




.. GENERATED FROM PYTHON SOURCE LINES 22-24

Below and above the threshold
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-34

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    for b in (1.7, 3.0):
        traj = Brusselator([1.2, 1.0], a=a, b=b).integrate((0.0, 60.0), dt=1e-2, method="rk4")
        axes[0].plot(traj.y[:, 0], traj.y[:, 1], lw=1, label=f"b={b}")
    axes[0].set_xlabel("x")
    axes[0].set_ylabel("y")
    axes[0].set_title("Spiral into the fixed point vs. limit cycle")
    axes[0].legend()




.. image-sg:: /api/gallery/ode_dynamics/limit_cycles/images/sphx_glr_plot_03_brusselator_hopf_001.png
   :alt: Spiral into the fixed point vs. limit cycle
   :srcset: /api/gallery/ode_dynamics/limit_cycles/images/sphx_glr_plot_03_brusselator_hopf_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11aeeb380>



.. GENERATED FROM PYTHON SOURCE LINES 35-37

Oscillation amplitude across the bifurcation
--------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 37-52

.. code-block:: Python


    bs = np.linspace(1.5, 3.5, 21)
    amps = []
    for b in bs:
        traj = Brusselator([1.2, 1.0], a=a, b=b).integrate((0.0, 300.0), dt=1e-2, method="rk4")
        amps.append(np.ptp(traj.y[-5000:, 0]))
    axes[1].plot(bs, amps, "o-")
    axes[1].axvline(b_c, color="gray", ls="--", label="b_c = 1 + a^2")
    axes[1].set_xlabel("b")
    axes[1].set_ylabel("peak-to-peak x")
    axes[1].set_title("Oscillations switch on at b_c")
    axes[1].legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 5.034 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_limit_cycles_plot_03_brusselator_hopf.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_brusselator_hopf.ipynb <plot_03_brusselator_hopf.ipynb>`

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

      :download:`Download Python source code: plot_03_brusselator_hopf.py <plot_03_brusselator_hopf.py>`

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

      :download:`Download zipped: plot_03_brusselator_hopf.zip <plot_03_brusselator_hopf.zip>`


.. only:: html

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

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