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

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

.. _sphx_glr_api_gallery_chaos_continuous_systems_plot_lorenz_attractor.py:


Lorenz Attractor
================

The Lorenz system is the archetypal chaotic flow: three coupled nonlinear
autonomous ODEs, originally derived as a drastic truncation of the equations
for atmospheric (Rayleigh-Benard) convection,

.. math::

    \dot{x} &= \sigma (y - x) \\
    \dot{y} &= x (\rho - z) - y \\
    \dot{z} &= x y - \beta z

whose trajectories, for the classic parameters :math:`\sigma=10`,
:math:`\rho=28`, :math:`\beta=8/3` used below, settle onto a butterfly-shaped
strange attractor rather than a fixed point or limit cycle. This example
integrates a trajectory with
:meth:`physicskit.chaos.systems.continuous.Lorenz.trajectory` (the
Numba-accelerated RK4 integrator) and plots it in 3D.

.. GENERATED FROM PYTHON SOURCE LINES 22-30

.. code-block:: Python


    import matplotlib.pyplot as plt

    from physicskit.chaos.systems.continuous import Lorenz
    from physicskit.chaos.visualizers.section import plot_poincare_map

    system = Lorenz(sigma=10.0, rho=28.0, beta=8.0 / 3.0)








.. GENERATED FROM PYTHON SOURCE LINES 31-35

Integrate
---------
We discard a short initial transient so the plotted trajectory starts
already on the attractor.

.. GENERATED FROM PYTHON SOURCE LINES 35-38

.. code-block:: Python

    t, states = system.trajectory(n_steps=20000, dt=0.01)
    states = states[500:]








.. GENERATED FROM PYTHON SOURCE LINES 39-41

Plot
----

.. GENERATED FROM PYTHON SOURCE LINES 41-51

.. code-block:: Python

    fig = plt.figure(figsize=(7, 6))
    ax = fig.add_subplot(projection="3d")
    ax.plot(states[:, 0], states[:, 1], states[:, 2], lw=0.4, color="darkorange")
    ax.set_xlabel("x")
    ax.set_ylabel("y")
    ax.set_zlabel("z")
    ax.set_title("Lorenz attractor")

    plt.show()




.. image-sg:: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_lorenz_attractor_001.png
   :alt: Lorenz attractor
   :srcset: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_lorenz_attractor_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 52-63

Poincare section: slicing the butterfly
--------------------------------------------
The tangled 3D curve above is far easier to read one slice at a time.
:func:`~physicskit.chaos.visualizers.section.plot_poincare_map` samples the
flow only at the instants it pierces the plane :math:`z = \rho - 1 = 27`
(through the two unstable fixed points the attractor's two lobes wind
around) while ascending (:math:`\dot{z} > 0`), and scatters each crossing's
:math:`(x, y)`. Rather than a tangled ribbon, the result is two crisp,
curved bands -- one per lobe -- whose thickness reveals the attractor's
fine, sheet-like layered structure directly, without the 3D projection's
self-occlusion.

.. GENERATED FROM PYTHON SOURCE LINES 63-78

.. code-block:: Python

    fig2, ax2 = plot_poincare_map(
        system,
        system.initial_state(),
        coord=2,
        value=system.rho - 1.0,
        direction=1.0,
        plot_coords=(0, 1),
        t_max=200.0,
        dt=0.005,
    )
    ax2.set_xlabel("x")
    ax2.set_ylabel("y")
    ax2.set_title(rf"Lorenz Poincare section at $z = \rho - 1 = {system.rho - 1.0:g}$")

    plt.show()



.. image-sg:: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_lorenz_attractor_002.png
   :alt: Lorenz Poincare section at $z = \rho - 1 = 27$
   :srcset: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_lorenz_attractor_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.071 seconds)


.. _sphx_glr_download_api_gallery_chaos_continuous_systems_plot_lorenz_attractor.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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