.. 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_chua_circuit.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_chua_circuit.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_chaos_continuous_systems_plot_chua_circuit.py:


Chua's Circuit: the Double-Scroll Attractor
==============================================

Chua's circuit is built from just a handful of standard electronic
components -- two capacitors, one inductor, one resistor, and a single
piecewise-linear nonlinear resistor (the "Chua diode") -- yet it is chaotic.
In dimensionless variables :math:`(x, y, z)` (proportional to the two
capacitor voltages and the inductor current), its dynamics are

.. math::

    \dot{x} &= \alpha (y - x - h(x)) \\
    \dot{y} &= x - y + z \\
    \dot{z} &= -\beta y \\
    h(x) &= m_1 x + \tfrac{1}{2}(m_0 - m_1)\left(|x + 1| - |x - 1|\right)

where :math:`h(x)` is the Chua diode's piecewise-linear current-voltage
characteristic (slope :math:`m_0` near the origin, :math:`m_1` for large
:math:`|x|`), and :math:`\alpha`, :math:`\beta` are ratios of the circuit's
capacitances and inductance. It holds a special place in chaos theory as the
first system whose chaotic behavior was confirmed both by simulation and by
direct physical experiment in real hardware, closing the "is chaos just a
numerical artifact?" question of the 1980s. For the classic parameters
(:math:`\alpha=15.6`, :math:`\beta=28`, :math:`m_0=-8/7`, :math:`m_1=-5/7`)
it produces the famous *double-scroll* attractor. Alongside the attractor
itself, :func:`physicskit.chaos.visualizers.divergence.plot_lyapunov_divergence`
quantifies the "chaotic" claim directly, by tracking how fast two initially
nearby trajectories separate.

.. GENERATED FROM PYTHON SOURCE LINES 31-39

.. code-block:: Python


    import matplotlib.pyplot as plt

    from physicskit.chaos.systems.continuous import Chua
    from physicskit.chaos.visualizers.divergence import plot_lyapunov_divergence

    system = Chua(alpha=15.6, beta=28.0)








.. GENERATED FROM PYTHON SOURCE LINES 40-42

Integrate
---------

.. GENERATED FROM PYTHON SOURCE LINES 42-45

.. code-block:: Python

    t, states = system.trajectory(n_steps=40000, dt=0.01)
    states = states[2000:]








.. GENERATED FROM PYTHON SOURCE LINES 46-54

The double scroll, and the divergence that makes it chaotic
-----------------------------------------------------------------
The trajectory spirals outward on one lobe, crosses over, spirals outward
on the other, and back again -- unpredictably, forever. Alongside it,
:func:`~physicskit.chaos.visualizers.divergence.plot_lyapunov_divergence`
makes the "chaotic" claim quantitative: two trajectories launched an
infinitesimal distance apart separate exponentially, at a rate
:math:`\lambda_{\max} > 0`.

.. GENERATED FROM PYTHON SOURCE LINES 54-68

.. code-block:: Python

    fig = plt.figure(figsize=(13, 6))
    ax = fig.add_subplot(1, 2, 1, projection="3d")
    ax.plot(states[:, 0], states[:, 1], states[:, 2], lw=0.3, color="mediumvioletred")
    ax.set_xlabel("x")
    ax.set_ylabel("y")
    ax.set_zlabel("z")
    ax.set_title("Chua's circuit: double-scroll attractor")

    ax_div = fig.add_subplot(1, 2, 2)
    _, _, lam = plot_lyapunov_divergence(system, t_max=20.0, n_points=1500, ax=ax_div, seed=0)
    ax_div.set_title(f"Trajectory divergence: $\\lambda_{{max}} \\approx {lam:.3f} > 0$")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_chua_circuit_001.png
   :alt: Chua's circuit: double-scroll attractor, Trajectory divergence: $\lambda_{max} \approx 0.706 > 0$
   :srcset: /api/gallery/chaos/continuous_systems/images/sphx_glr_plot_chua_circuit_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.390 seconds)


.. _sphx_glr_download_api_gallery_chaos_continuous_systems_plot_chua_circuit.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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