.. _sphx_glr_api_gallery_ode_dynamics_limit_cycles:

Limit cycles
------------

The Van der Pol oscillator's amplitude-independent limit cycle.


.. 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="Trajectories starting both inside and outside the eventual limit cycle converge onto the same closed orbit -- the defining property that distinguishes a limit cycle from a linear center (whose orbit amplitude is set entirely by the initial condition).">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/limit_cycles/images/thumb/sphx_glr_plot_01_van_der_pol_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/limit_cycles/plot_01_van_der_pol`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Van der Pol limit cycle</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="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 \mu(1 - x^2) changes sign at |x| = 1 -- and indeed its limit cycle must cross those lines.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/limit_cycles/images/thumb/sphx_glr_plot_02_bendixson_criterion_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/limit_cycles/plot_02_bendixson_criterion`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Bendixson's negative criterion</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The Brusselator&#x27;s fixed point (a, b/a) is stable until b crosses 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.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/limit_cycles/images/thumb/sphx_glr_plot_03_brusselator_hopf_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/limit_cycles/plot_03_brusselator_hopf`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Brusselator's Hopf bifurcation</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/ode_dynamics/limit_cycles/plot_01_van_der_pol
   /api/gallery/ode_dynamics/limit_cycles/plot_02_bendixson_criterion
   /api/gallery/ode_dynamics/limit_cycles/plot_03_brusselator_hopf