.. _sphx_glr_api_gallery_kinetics_oscillators:

Oscillating reactions
-----------------------

Chemical mechanisms that oscillate instead of relaxing to equilibrium:
the Brusselator limit cycle, Lotka's autocatalytic scheme with its
neutral orbits, and the Oregonator model of the Belousov-Zhabotinsky
reaction.


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


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

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    <div class="sphx-glr-thumbcontainer" tooltip="Below the Hopf-bifurcation threshold (:math:`B &lt; 1 + A^2`) the Brusselator&#x27;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 X^2 Y term.">

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  .. image:: /api/gallery/kinetics/oscillators/images/thumb/sphx_glr_plot_01_brusselator_thumb.png
    :alt:

  :doc:`/api/gallery/kinetics/oscillators/plot_01_brusselator`

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      <div class="sphx-glr-thumbnail-title">Prigogine and Lefever's Brusselator limit cycle</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Lotka&#x27;s 1920 chemical scheme, with the reservoir A held constant,">

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  .. image:: /api/gallery/kinetics/oscillators/images/thumb/sphx_glr_plot_02_lotka_autocatalytic_oscillator_thumb.png
    :alt:

  :doc:`/api/gallery/kinetics/oscillators/plot_02_lotka_autocatalytic_oscillator`

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      <div class="sphx-glr-thumbnail-title">Lotka's autocatalytic oscillator and its neutral orbits</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Field and Noyes (1974) boiled the FKN mechanism of the Belousov-Zhabotinsky (BZ) reaction -- cerium-catalyzed oxidation of malonic acid by bromate -- down to three intermediates: HBrO\ 2 (x), bromide (y), and the oxidized catalyst Ce(IV) (z). The resulting &quot;Oregonator&quot; (~chemistrykit.kinetics.systems.oscillators.Oregonator) produces the BZ reaction&#x27;s hallmark relaxation oscillations: long quiet stretches while bromide is slowly consumed, then an autocatalytic burst of HBrO\ 2 that oxidizes the catalyst (the colour flash seen by Belousov), which in turn regenerates bromide and shuts the burst off. Raising the stoichiometric factor f well above 2 stabilizes the steady state and the oscillation disappears.">

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  .. image:: /api/gallery/kinetics/oscillators/images/thumb/sphx_glr_plot_03_oregonator_bz_reaction_thumb.png
    :alt:

  :doc:`/api/gallery/kinetics/oscillators/plot_03_oregonator_bz_reaction`

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      <div class="sphx-glr-thumbnail-title">The Oregonator: oscillations of the Belousov-Zhabotinsky reaction</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/kinetics/oscillators/plot_01_brusselator
   /api/gallery/kinetics/oscillators/plot_02_lotka_autocatalytic_oscillator
   /api/gallery/kinetics/oscillators/plot_03_oregonator_bz_reaction