.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/ode_dynamics/stability/plot_02_lyapunov_function.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_ode_dynamics_stability_plot_02_lyapunov_function.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_ode_dynamics_stability_plot_02_lyapunov_function.py:


Lyapunov's direct method
==============================

For a stable linear system :math:`\dot x = Ax`, solving the Lyapunov
equation :math:`A^T P + PA = -Q` gives an energy-like function
:math:`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.

.. GENERATED FROM PYTHON SOURCE LINES 13-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.ode_dynamics.systems.phase_portrait import Linear2D
    from mathematicskit.ode_dynamics.systems.stability import lyapunov_quadratic_form

    A = np.array([[-0.3, 2.0], [-1.0, -0.3]])
    res = lyapunov_quadratic_form(A)
    print("P =\n", np.round(res.P, 4))
    print("P positive definite (origin asymptotically stable):", res.positive_definite)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    P =
     [[1.2679 0.1196]
     [0.1196 2.4641]]
    P positive definite (origin asymptotically stable): True




.. GENERATED FROM PYTHON SOURCE LINES 25-27

Trajectories cross the level sets of V inward
---------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-45

.. code-block:: Python


    xs = np.linspace(-2.5, 2.5, 200)
    X, Y = np.meshgrid(xs, xs)
    V = res.P[0, 0] * X**2 + 2 * res.P[0, 1] * X * Y + res.P[1, 1] * Y**2
    traj = Linear2D([2.0, 1.0], A=A).integrate((0.0, 20.0), dt=1e-2, method="rk4")
    V_traj = np.einsum("ti,ij,tj->t", traj.y, res.P, traj.y)

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    axes[0].contour(X, Y, V, levels=12, cmap="viridis")
    axes[0].plot(traj.y[:, 0], traj.y[:, 1], "k", lw=1)
    axes[0].set_aspect("equal")
    axes[0].set_title("Level sets of V(x) = x^T P x and a trajectory")
    axes[1].semilogy(traj.t, V_traj)
    axes[1].set_xlabel("t")
    axes[1].set_ylabel("V(x(t))")
    axes[1].set_title("V decreases monotonically along the flow")
    fig.tight_layout()




.. image-sg:: /api/gallery/ode_dynamics/stability/images/sphx_glr_plot_02_lyapunov_function_001.png
   :alt: Level sets of V(x) = x^T P x and a trajectory, V decreases monotonically along the flow
   :srcset: /api/gallery/ode_dynamics/stability/images/sphx_glr_plot_02_lyapunov_function_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 46-48

An unstable matrix admits no positive-definite P
-------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 48-52

.. code-block:: Python


    print("unstable spiral, P positive definite:", lyapunov_quadratic_form([[0.1, 1.0], [-1.0, 0.1]]).positive_definite)

    plt.show()




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    unstable spiral, P positive definite: False





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.330 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_stability_plot_02_lyapunov_function.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_lyapunov_function.ipynb <plot_02_lyapunov_function.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_lyapunov_function.py <plot_02_lyapunov_function.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_lyapunov_function.zip <plot_02_lyapunov_function.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_