.. _sphx_glr_api_gallery_chaos_bifurcations:

Bifurcations
------------

Examples for :mod:`physicskit.chaos.systems.bifurcations`: the Hopf
bifurcation, where a fixed point gives birth to a limit cycle, in its normal
form and in the Brusselator chemical oscillator.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="When a pair of complex eigenvalues \mu \pm i\omega of a fixed point crosses the imaginary axis, the fixed point turns from a stable into an unstable spiral and a limit cycle appears (Andronov 1929, Hopf 1942). This is how oscillations start in systems as different as chemical reactions, lasers, heart cells and flow past a cylinder, and it is a different route from the period-doubling cascade: there is no sequence of bifurcations, just one cycle whose amplitude grows from zero. Near onset every such system reduces to the normal form">

.. only:: html

  .. image:: /api/gallery/chaos/bifurcations/images/thumb/sphx_glr_plot_hopf_bifurcation_thumb.png
    :alt:

  :doc:`/api/gallery/chaos/bifurcations/plot_hopf_bifurcation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Hopf bifurcation: a limit cycle born from a fixed point</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/chaos/bifurcations/plot_hopf_bifurcation