.. _sphx_glr_api_gallery_ode_dynamics_stability:

Fixed-point stability
----------------------

Jacobian linearization and the trace-determinant classification of
node/saddle/spiral/center fixed points.


.. 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="Poincaré&#x27;s qualitative theory sorts a planar fixed point into one of four types -- node, saddle, spiral (focus), or center -- using only the eigenvalues of the linearized flow, without solving the equations. The eigenvalues depend only on the Jacobian&#x27;s trace \tau and determinant \Delta, so every type occupies its own region of the (\tau, \Delta) plane. This script classifies six canonical matrices, draws their phase portraits, and places each one on Poincaré&#x27;s trace-determinant diagram.">

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  .. image:: /api/gallery/ode_dynamics/stability/images/thumb/sphx_glr_plot_01_classification_zoo_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/stability/plot_01_classification_zoo`

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      <div class="sphx-glr-thumbnail-title">Poincaré's classification of fixed points: nodes, saddles, spirals, centers</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="For a stable linear system \dot x = Ax, solving the Lyapunov equation A^T P + PA = -Q gives an energy-like function V(x) = x^T P x whose level sets are ellipses that every trajectory crosses inward -- a proof of stability that never solves the ODE itself.">

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  .. image:: /api/gallery/ode_dynamics/stability/images/thumb/sphx_glr_plot_02_lyapunov_function_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/stability/plot_02_lyapunov_function`

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      <div class="sphx-glr-thumbnail-title">Lyapunov's direct method</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/ode_dynamics/stability/plot_01_classification_zoo
   /api/gallery/ode_dynamics/stability/plot_02_lyapunov_function