.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/ode_dynamics/chaotic_flows/plot_02_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_ode_dynamics_chaotic_flows_plot_02_rossler_attractor.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_ode_dynamics_chaotic_flows_plot_02_rossler_attractor.py:


The Rossler attractor
===========================

Rossler's flow has a single nonlinear term, yet for :math:`a = b =
0.2`, :math:`c = 5.7` it settles onto a chaotic attractor: orbits spiral
outward near the plane :math:`z \approx 0`, are lifted and folded back
when :math:`x` exceeds :math:`c`, and never exactly repeat. Plotting
each loop's maximum of :math:`x` against the previous one gives a
Lorenz-style return map, a thin single-humped curve.

.. GENERATED FROM PYTHON SOURCE LINES 14-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.ode_dynamics.systems.chaotic_flows import RosslerSystem, rossler_fixed_points

    print("fixed points:\n", np.round(rossler_fixed_points(), 4))

    system = RosslerSystem([1.0, 1.0, 0.0])
    system.integrate((0.0, 100.0), dt=1e-2, method="rk4")
    traj = system.integrate((100.0, 1100.0), dt=1e-2, method="rk4")
    x, y, z = traj.y.T





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

 .. code-block:: none

    fixed points:
     [[ 7.00000e-03 -3.51000e-02  3.51000e-02]
     [ 5.69300e+00 -2.84649e+01  2.84649e+01]]




.. GENERATED FROM PYTHON SOURCE LINES 27-29

The attractor and its return map
--------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-46

.. code-block:: Python


    peaks = x[1:-1][(x[1:-1] > x[:-2]) & (x[1:-1] > x[2:])]
    fig = plt.figure(figsize=(11, 4.5))
    ax3d = fig.add_subplot(1, 2, 1, projection="3d")
    ax3d.plot(x, y, z, lw=0.3)
    ax3d.set_xlabel("x")
    ax3d.set_ylabel("y")
    ax3d.set_zlabel("z")
    ax3d.set_title("Rossler attractor (a=b=0.2, c=5.7)")
    ax = fig.add_subplot(1, 2, 2)
    ax.plot(peaks[:-1], peaks[1:], ".", ms=3)
    ax.set_xlabel("x_max (n)")
    ax.set_ylabel("x_max (n+1)")
    ax.set_title("Return map of successive maxima")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/ode_dynamics/chaotic_flows/images/sphx_glr_plot_02_rossler_attractor_001.png
   :alt: Rossler attractor (a=b=0.2, c=5.7), Return map of successive maxima
   :srcset: /api/gallery/ode_dynamics/chaotic_flows/images/sphx_glr_plot_02_rossler_attractor_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.309 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_chaotic_flows_plot_02_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_02_rossler_attractor.ipynb <plot_02_rossler_attractor.ipynb>`

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

      :download:`Download Python source code: plot_02_rossler_attractor.py <plot_02_rossler_attractor.py>`

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

      :download:`Download zipped: plot_02_rossler_attractor.zip <plot_02_rossler_attractor.zip>`


.. only:: html

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

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