.. 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_rossler_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_rossler_attractor.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_chaos_continuous_systems_plot_rossler_attractor.py:


Rossler Attractor
=================

The Rossler system is three coupled nonlinear autonomous ODEs, deliberately
designed to be simpler than Lorenz while still chaotic:

.. math::

    \dot{x} &= -y - z \\
    \dot{y} &= x + a y \\
    \dot{z} &= b + z (x - c)

It produces a simpler-looking, ribbon-like strange attractor than Lorenz,
built from a single stretch-and-fold mechanism: trajectories spiral
outward in the :math:`(x, y)` plane (driven by the linear :math:`-y-z` and
:math:`x+ay` terms) until :math:`x` exceeds :math:`c`, at which point the
:math:`z (x - c)` term kicks the trajectory up and folds it back down onto
the spiral. This example integrates a trajectory with
:meth:`physicskit.chaos.systems.continuous.Rossler.trajectory` and plots it
in 3D, using the classic chaotic parameters :math:`a=0.2`, :math:`b=0.2`,
:math:`c=5.7`.

.. GENERATED FROM PYTHON SOURCE LINES 24-32

.. code-block:: Python


    import matplotlib.pyplot as plt

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

    system = Rossler(a=0.2, b=0.2, c=5.7)








.. GENERATED FROM PYTHON SOURCE LINES 33-35

Integrate
---------

.. GENERATED FROM PYTHON SOURCE LINES 35-38

.. code-block:: Python

    t, states = system.trajectory(n_steps=40000, dt=0.02)
    states = states[1000:]








.. 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="teal")
    ax.set_xlabel("x")
    ax.set_ylabel("y")
    ax.set_zlabel("z")
    ax.set_title("Rossler attractor")

    plt.show()




.. image-sg:: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_rossler_attractor_001.png
   :alt: Rossler attractor
   :srcset: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_rossler_attractor_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 52-63

Poincare section: the stretch-and-fold mechanism, laid bare
------------------------------------------------------------------
Sampling the flow only at the instants it crosses the plane :math:`y = 0`
moving outward (:math:`\dot{y} > 0`) -- the natural section for a spiral
that winds around the :math:`z`-axis -- turns the ribbon-like 3D attractor
into a near-one-dimensional curve of successive crossings'
:math:`(x, z)`. That the points collapse almost onto a single folded curve
(rather than filling a 2D patch) is the direct visual signature of the
single stretch-and-fold mechanism described above: each pass around the
spiral maps an interval of :math:`x` values into a slightly different,
folded-over interval, iteration after iteration.

.. GENERATED FROM PYTHON SOURCE LINES 63-79

.. code-block:: Python

    fig2, ax2 = plot_poincare_map(
        system,
        system.initial_state(),
        coord=1,
        value=0.0,
        direction=1.0,
        plot_coords=(0, 2),
        t_max=400.0,
        dt=0.005,
        s=4,
    )
    ax2.set_xlabel("x")
    ax2.set_ylabel("z")
    ax2.set_title("Rossler Poincare section at y = 0 (ascending)")

    plt.show()



.. image-sg:: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_rossler_attractor_002.png
   :alt: Rossler Poincare section at y = 0 (ascending)
   :srcset: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_rossler_attractor_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.084 seconds)


.. _sphx_glr_download_api_gallery_chaos_continuous_systems_plot_rossler_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_rossler_attractor.ipynb <plot_rossler_attractor.ipynb>`

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

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

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

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


.. only:: html

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

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