.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/chaos/chaos_metrics/plot_lyapunov_divergence.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_chaos_chaos_metrics_plot_lyapunov_divergence.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_chaos_chaos_metrics_plot_lyapunov_divergence.py:


Lyapunov Divergence: Chaotic vs. Non-Chaotic
==============================================

:func:`physicskit.chaos.visualizers.divergence.plot_lyapunov_divergence` integrates
two trajectories starting an infinitesimal distance :math:`\delta_0` apart
and tracks :math:`\ln(\delta_t / \delta_0)` over time, where
:math:`\delta_t` is their separation at time :math:`t`. For chaotic motion
this separation grows exponentially, :math:`\delta_t \sim \delta_0
e^{\lambda_{\max} t}`, so a straight, positively-sloped line (its slope the
largest Lyapunov exponent :math:`\lambda_{\max} > 0`) is the signature of
chaos; a system that is *not* chaotic instead shows the separation shrinking
(:math:`\lambda_{\max} < 0`) as both trajectories relax toward the same
attracting state. Two systems are contrasted below: the chaotic Lorenz
attractor, :math:`\dot{x}=\sigma(y-x)`, :math:`\dot{y}=x(\rho-z)-y`,
:math:`\dot{z}=xy-\beta z`; and the Duffing oscillator,
:math:`\ddot{x}+\delta_D \dot{x}+\alpha x+\beta_D x^3=\gamma\cos(\omega t)`,
with its cubic term and forcing switched off (:math:`\beta_D=\gamma=0`,
:math:`\alpha=1 > 0`), which reduces it to a plain damped linear
oscillator, :math:`\ddot{x}+\delta_D \dot{x}+x=0`: every trajectory relaxes
to the same fixed point, so nearby trajectories converge rather than
diverge.

.. GENERATED FROM PYTHON SOURCE LINES 24-30

.. code-block:: Python


    import matplotlib.pyplot as plt

    from physicskit.chaos.systems.continuous import Duffing, Lorenz
    from physicskit.chaos.visualizers.divergence import plot_lyapunov_divergence








.. GENERATED FROM PYTHON SOURCE LINES 31-33

Chaotic case: Lorenz
---------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-35

.. code-block:: Python

    lorenz = Lorenz()








.. GENERATED FROM PYTHON SOURCE LINES 36-42

Non-chaotic case: an undriven, damped (linear) oscillator
------------------------------------------------------------
Setting the Duffing oscillator's forcing amplitude to zero and its cubic
term to zero leaves a plain damped harmonic oscillator: every trajectory
relaxes to the same fixed point, so nearby trajectories converge rather
than diverge.

.. GENERATED FROM PYTHON SOURCE LINES 42-52

.. code-block:: Python

    damped = Duffing(delta=0.3, alpha=1.0, beta=0.0, gamma=0.0, omega=1.0)

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(13, 5))
    _, _, lam_lorenz = plot_lyapunov_divergence(lorenz, t_max=20.0, n_points=1000, ax=ax1, seed=0)
    _, _, lam_damped = plot_lyapunov_divergence(damped, t_max=20.0, n_points=1000, ax=ax2, seed=0)
    ax1.set_title(f"Lorenz (chaotic): $\\lambda_{{max}} \\approx {lam_lorenz:.3f} > 0$")
    ax2.set_title(f"Damped oscillator: $\\lambda_{{max}} \\approx {lam_damped:.3f} < 0$")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/chaos/chaos_metrics/images/sphx_glr_plot_lyapunov_divergence_001.png
   :alt: Lorenz (chaotic): $\lambda_{max} \approx 0.260 > 0$, Damped oscillator: $\lambda_{max} \approx -0.148 < 0$
   :srcset: /api/gallery/chaos/chaos_metrics/images/sphx_glr_plot_lyapunov_divergence_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.217 seconds)


.. _sphx_glr_download_api_gallery_chaos_chaos_metrics_plot_lyapunov_divergence.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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