.. _sphx_glr_api_gallery_ode_dynamics_chaotic_flows:

Chaotic flows
-------------

Lorenz's butterfly attractor and its sensitive dependence on initial
conditions, and the Rossler attractor, a minimal three-dimensional
chaotic flow.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Lorenz&#x27;s 1963 convection model \dot x = \sigma(y - x), \dot y = x(\rho - z) - y, \dot z = xy - \beta z is fully deterministic, yet for \sigma = 10, \rho = 28, \beta = 8/3 its orbits wind forever around a butterfly-shaped strange attractor. This script repeats Lorenz&#x27;s accidental experiment: restart a run from a state rounded to three decimal places, and watch the two forecasts agree for a while, then diverge completely.">

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  .. image:: /api/gallery/ode_dynamics/chaotic_flows/images/thumb/sphx_glr_plot_01_lorenz_butterfly_effect_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/chaotic_flows/plot_01_lorenz_butterfly_effect`

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      <div class="sphx-glr-thumbnail-title">Lorenz's butterfly effect: deterministic chaos</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Rossler&#x27;s flow has a single nonlinear term, yet for a = b = 0.2, c = 5.7 it settles onto a chaotic attractor: orbits spiral outward near the plane z \approx 0, are lifted and folded back when x exceeds c, and never exactly repeat. Plotting each loop&#x27;s maximum of x against the previous one gives a Lorenz-style return map, a thin single-humped curve.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/chaotic_flows/images/thumb/sphx_glr_plot_02_rossler_attractor_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/chaotic_flows/plot_02_rossler_attractor`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Rossler attractor</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/ode_dynamics/chaotic_flows/plot_01_lorenz_butterfly_effect
   /api/gallery/ode_dynamics/chaotic_flows/plot_02_rossler_attractor