.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/ode_dynamics/bifurcations/plot_01_normal_forms.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_bifurcations_plot_01_normal_forms.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_ode_dynamics_bifurcations_plot_01_normal_forms.py:


Saddle-node, pitchfork, and Hopf bifurcation diagrams
==========================================================

Plots the closed-form fixed points (or limit-cycle radius) of the three
canonical bifurcation normal forms as a function of the control
parameter ``r``, showing the characteristic collision-and-annihilation,
splitting, and birth-of-a-limit-cycle shapes.

.. GENERATED FROM PYTHON SOURCE LINES 12-17

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.ode_dynamics.systems.bifurcations import hopf_limit_cycle_radius, pitchfork_fixed_points, saddle_node_fixed_points








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Saddle-node: two fixed points collide and annihilate
------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-24

.. code-block:: Python


    r_sn = np.linspace(-4.0, 1.0, 400)
    fp_sn = saddle_node_fixed_points(r_sn)








.. GENERATED FROM PYTHON SOURCE LINES 25-27

Pitchfork: one branch splits into three
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-31

.. code-block:: Python


    r_pf = np.linspace(-4.0, 4.0, 400)
    fp_pf = pitchfork_fixed_points(r_pf, kind="supercritical")








.. GENERATED FROM PYTHON SOURCE LINES 32-34

Hopf: a limit cycle is born and grows
--------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-56

.. code-block:: Python


    r_hopf = np.linspace(-2.0, 4.0, 400)
    radius_hopf = hopf_limit_cycle_radius(r_hopf)

    fig, axes = plt.subplots(1, 3, figsize=(13, 4))
    axes[0].plot(r_sn, fp_sn[:, 0], color="steelblue")
    axes[0].plot(r_sn, fp_sn[:, 1], color="firebrick", linestyle="--")
    axes[0].set_title("Saddle-node")
    axes[0].set_xlabel("r")

    axes[1].plot(r_pf, fp_pf[:, 0], color="black")
    axes[1].plot(r_pf, fp_pf[:, 1], color="steelblue")
    axes[1].plot(r_pf, fp_pf[:, 2], color="steelblue")
    axes[1].set_title("Pitchfork (supercritical)")
    axes[1].set_xlabel("r")

    axes[2].plot(r_hopf, radius_hopf, color="steelblue")
    axes[2].set_title("Hopf: limit-cycle radius")
    axes[2].set_xlabel("r")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/ode_dynamics/bifurcations/images/sphx_glr_plot_01_normal_forms_001.png
   :alt: Saddle-node, Pitchfork (supercritical), Hopf: limit-cycle radius
   :srcset: /api/gallery/ode_dynamics/bifurcations/images/sphx_glr_plot_01_normal_forms_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.079 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_bifurcations_plot_01_normal_forms.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_normal_forms.ipynb <plot_01_normal_forms.ipynb>`

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

      :download:`Download Python source code: plot_01_normal_forms.py <plot_01_normal_forms.py>`

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

      :download:`Download zipped: plot_01_normal_forms.zip <plot_01_normal_forms.zip>`


.. only:: html

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

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