.. _sphx_glr_api_gallery_ode_dynamics_phase_portrait:

Phase portraits
---------------

Linear and nonlinear planar flows and their vector fields.


.. 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="The nonlinear pendulum \ddot\theta = -\sin\theta has no elementary closed-form solution, but Poincaré&#x27;s qualitative approach does not need one. Locate the fixed points, classify each from its linearization, and sketch the vector field: the global picture of librating orbits around the centers, rotating orbits above and below, and separatrices joining the saddles follows without solving anything. Integrated trajectories are overlaid only to confirm the prediction.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/phase_portrait/images/thumb/sphx_glr_plot_01_vector_fields_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/phase_portrait/plot_01_vector_fields`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Poincaré's qualitative method: the pendulum's phase portrait</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/ode_dynamics/phase_portrait/plot_01_vector_fields