.. _sphx_glr_api_gallery_chaos_continuous_systems:

Continuous Systems
------------------

Examples illustrating the continuous-time chaotic flows in
:mod:`physicskit.chaos.systems.continuous`: the Lorenz and Rossler attractors, the double pendulum, the
forced/damped Duffing oscillator, Chua's circuit (the double-scroll attractor), the restricted
three-body problem (with its five Lagrange points marked), the magnetic pendulum (fractal basins of
attraction), and the sinusoidally driven, damped pendulum (the period-doubling route to chaos). Each is
integrated with the Numba-accelerated RK4 integrator from :mod:`physicskit.chaos.core.integrators` via the
system's ``.trajectory()`` method.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Chua&#x27;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 &quot;Chua diode&quot;) -- yet it is chaotic. In dimensionless variables (x, y, z) (proportional to the two capacitor voltages and the inductor current), its dynamics are">

.. only:: html

  .. image:: /api/gallery/chaos/continuous_systems/images/thumb/sphx_glr_plot_chua_circuit_thumb.png
    :alt:

  :doc:`/api/gallery/chaos/continuous_systems/plot_chua_circuit`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Chua's Circuit: the Double-Scroll Attractor</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The double pendulum is a simple mechanical system -- two point masses m_1, m_2 on massless rigid rods of length l_1, l_2, hanging from a fixed pivot under gravity g, with no friction -- whose motion is nonetheless chaotic for large enough swings. In terms of the two rod angles \theta_1, \theta_2 from vertical and their angular velocities \omega_1 = \dot\theta_1, \omega_2 = \dot\theta_2, the equations of motion are">

.. only:: html

  .. image:: /api/gallery/chaos/continuous_systems/images/thumb/sphx_glr_plot_double_pendulum_thumb.gif
    :alt:

  :doc:`/api/gallery/chaos/continuous_systems/plot_double_pendulum`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Double Pendulum</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The sinusoidally forced, damped pendulum obeys">

.. only:: html

  .. image:: /api/gallery/chaos/continuous_systems/images/thumb/sphx_glr_plot_driven_pendulum_thumb.png
    :alt:

  :doc:`/api/gallery/chaos/continuous_systems/plot_driven_pendulum`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Driven Pendulum: the Period-Doubling Route to Chaos</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The forced, damped Duffing oscillator models a particle in a double-well potential driven by a periodic force,">

.. only:: html

  .. image:: /api/gallery/chaos/continuous_systems/images/thumb/sphx_glr_plot_duffing_oscillator_thumb.png
    :alt:

  :doc:`/api/gallery/chaos/continuous_systems/plot_duffing_oscillator`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Duffing Oscillator</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="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,">

.. only:: html

  .. image:: /api/gallery/chaos/continuous_systems/images/thumb/sphx_glr_plot_lorenz_attractor_thumb.png
    :alt:

  :doc:`/api/gallery/chaos/continuous_systems/plot_lorenz_attractor`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Lorenz Attractor</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The magnetic pendulum is a damped bob, modeled in the small-swing (flat, 2D) limit at position (x, y), pulled toward the origin by a linear restoring force and toward each of N fixed magnets at (m_{x,k}, m_{y,k}) by a softened inverse-square force:">

.. only:: html

  .. image:: /api/gallery/chaos/continuous_systems/images/thumb/sphx_glr_plot_magnetic_pendulum_thumb.png
    :alt:

  :doc:`/api/gallery/chaos/continuous_systems/plot_magnetic_pendulum`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Magnetic Pendulum: Fractal Basins of Attraction</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The planar circular restricted three-body problem (CR3BP) places a massless third body under the gravity of two massive primaries -- masses 1-\mu and \mu in normalized units -- that are themselves in a fixed circular orbit about their common center of mass. In the rotating frame co-precessing with the primaries (which then sit fixed at (-\mu, 0) and (1-\mu, 0)), the third body&#x27;s position (x, y) obeys">

.. only:: html

  .. image:: /api/gallery/chaos/continuous_systems/images/thumb/sphx_glr_plot_restricted_three_body_thumb.gif
    :alt:

  :doc:`/api/gallery/chaos/continuous_systems/plot_restricted_three_body`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Restricted Three-Body Problem: Poincare's Original Chaos</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The Rossler system is three coupled nonlinear autonomous ODEs, deliberately designed to be simpler than Lorenz while still chaotic:">

.. only:: html

  .. image:: /api/gallery/chaos/continuous_systems/images/thumb/sphx_glr_plot_rossler_attractor_thumb.png
    :alt:

  :doc:`/api/gallery/chaos/continuous_systems/plot_rossler_attractor`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Rossler Attractor</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/chaos/continuous_systems/plot_chua_circuit
   /api/gallery/chaos/continuous_systems/plot_double_pendulum
   /api/gallery/chaos/continuous_systems/plot_driven_pendulum
   /api/gallery/chaos/continuous_systems/plot_duffing_oscillator
   /api/gallery/chaos/continuous_systems/plot_lorenz_attractor
   /api/gallery/chaos/continuous_systems/plot_magnetic_pendulum
   /api/gallery/chaos/continuous_systems/plot_restricted_three_body
   /api/gallery/chaos/continuous_systems/plot_rossler_attractor