.. _sphx_glr_api_gallery_ode_dynamics_logistic_map:

The logistic map
----------------

The period-doubling route to chaos and the Feigenbaum constant.


.. raw:: html

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


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

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="As the parameter r grows, the logistic map x \mapsto rx(1-x) doubles its period again and again -- 1, 2, 4, 8, ... -- at parameter values whose spacing shrinks by a constant factor, Feigenbaum&#x27;s \delta \approx 4.6692. Universality means the same \delta appears for any smooth one-humped map. This script measures \delta for the logistic map and for the unrelated sine map x \mapsto r\sin(\pi x) and shows both sequences of ratios converging to the one constant.">

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  .. image:: /api/gallery/ode_dynamics/logistic_map/images/thumb/sphx_glr_plot_01_bifurcation_cascade_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/logistic_map/plot_01_bifurcation_cascade`

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      <div class="sphx-glr-thumbnail-title">Feigenbaum's universal constant in the period-doubling cascade</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/ode_dynamics/logistic_map/plot_01_bifurcation_cascade