.. _sphx_glr_api_gallery_chaos_billiards:

Billiards
---------

Examples illustrating the five 2D billiard geometries in
:mod:`physicskit.chaos.systems.billiards`: a trajectory bouncing inside each shape,
and the boundary phase-space (Poincare) section that distinguishes
integrable geometries (Circle, Rectangle) from chaotic, defocusing ones
(Sinai, Bunimovich Stadium, Truncated Circle).


.. 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="A billiard particle moves in a straight line at constant speed inside a closed boundary and undergoes specular reflection, \mathbf{v}&#x27; = \mathbf{v} - 2(\mathbf{v}\cdot\mathbf{n})\,\mathbf{n}, whenever it strikes the boundary. The Bunimovich stadium&#x27;s boundary is two straight edges, y = \pm r for |x| \le a, joined by two semicircular caps of radius r centered at (\pm a, 0), where 2a is the straight-edge length. It is chaotic despite having no concave scatterer: the focusing semicircular caps defocus nearby trajectories after they refocus and diverge past the caps&#x27; centers of curvature (unlike a full circle, whose focusing never gets interrupted by a flat stretch). This example plots a single trajectory and the resulting Poincare section.">

.. only:: html

  .. image:: /api/gallery/chaos/billiards/images/thumb/sphx_glr_plot_bunimovich_stadium_thumb.png
    :alt:

  :doc:`/api/gallery/chaos/billiards/plot_bunimovich_stadium`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Bunimovich Stadium (Chaotic)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A billiard is a point particle moving in a straight line at constant speed inside a closed boundary, until it strikes the boundary and undergoes specular reflection -- the outgoing velocity \mathbf{v}&#x27; obtained from the incoming velocity \mathbf{v} and the local unit normal \mathbf{n} by">

.. only:: html

  .. image:: /api/gallery/chaos/billiards/images/thumb/sphx_glr_plot_circle_billiard_thumb.gif
    :alt:

  :doc:`/api/gallery/chaos/billiards/plot_circle_billiard`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Circle Billiard (Integrable)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A billiard particle moves in a straight line at constant speed inside a closed boundary and undergoes specular reflection, \mathbf{v}&#x27; = \mathbf{v} - 2(\mathbf{v}\cdot\mathbf{n})\,\mathbf{n}, whenever it strikes the boundary. Here the boundary is the ellipse">

.. only:: html

  .. image:: /api/gallery/chaos/billiards/images/thumb/sphx_glr_plot_ellipse_billiard_thumb.gif
    :alt:

  :doc:`/api/gallery/chaos/billiards/plot_ellipse_billiard`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Ellipse Billiard (Integrable, with Caustics)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A billiard particle moves in a straight line and reflects specularly off its boundary, \mathbf{v}&#x27; = \mathbf{v} - 2(\mathbf{v}\cdot\mathbf{n}) \,\mathbf{n}. The circular billiard (boundary x^2+y^2=r^2) is exactly integrable: \sin\phi -- the sine of the angle of incidence at each bounce -- is conserved forever, so two rays launched a hair&#x27;s breadth apart stay a hair&#x27;s breadth apart forever too. Slicing off even a tiny flat chord (turning it into a TruncatedCircleBilliard of boundary parameter cut) destroys this completely. This example launches two rays 10^{-8} radians apart into a circle truncated by a chord just 5% of the radius deep -- barely different from a perfect circle by eye -- and watches them track each other indistinguishably for dozens of bounces before a single grazing hit near the chord sends them down completely different paths.">

.. only:: html

  .. image:: /api/gallery/chaos/billiards/images/thumb/sphx_glr_plot_extreme_sensitivity_thumb.gif
    :alt:

  :doc:`/api/gallery/chaos/billiards/plot_extreme_sensitivity`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Extreme Sensitivity: a Slightly Truncated Circle</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Every billiard here is a point particle moving in a straight line at constant speed inside a closed boundary, undergoing specular reflection">

.. only:: html

  .. image:: /api/gallery/chaos/billiards/images/thumb/sphx_glr_plot_poincare_comparison_thumb.png
    :alt:

  :doc:`/api/gallery/chaos/billiards/plot_poincare_comparison`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Comparing All Five Billiards</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A billiard particle moves in a straight line at constant speed inside a closed boundary and undergoes specular reflection, \mathbf{v}&#x27; = \mathbf{v} - 2(\mathbf{v}\cdot\mathbf{n})\,\mathbf{n}, whenever it strikes the boundary. Here the boundary is the rectangle |x| \le w/2,\ |y| \le h/2 of width w and height h, whose walls are all axis-aligned, so \mathbf{n} is always \pm\hat{x} or \pm\hat{y} -- reflections off the horizontal and vertical walls therefore only ever flip the sign of one velocity component. The rectangular billiard is integrable: each individual ray only ever visits a handful of discrete \sin\phi values (the sine of the angle between the outgoing velocity and the local boundary tangent). This example plots a single trajectory and a Poincare section built from several rays launched at different angles.">

.. only:: html

  .. image:: /api/gallery/chaos/billiards/images/thumb/sphx_glr_plot_rectangle_billiard_thumb.gif
    :alt:

  :doc:`/api/gallery/chaos/billiards/plot_rectangle_billiard`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Rectangle Billiard (Integrable)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A billiard particle moves in a straight line at constant speed inside a closed boundary and undergoes specular reflection, \mathbf{v}&#x27; = \mathbf{v} - 2(\mathbf{v}\cdot\mathbf{n})\,\mathbf{n}, whenever it strikes the boundary. The Sinai billiard&#x27;s boundary is a square cell, |x| \le L/2,\ |y| \le L/2, with a circular scatterer of radius r_s &lt; L/2 removed from its center, x^2 + y^2 = r_s^2. It is a classic example of a defocusing chaotic billiard: because the scatterer is convex as seen from inside the cell, it disperses nearby trajectories exponentially fast (unlike the flat walls of a Rectangle billiard, which neither focus nor defocus). This example plots a single trajectory and the resulting Poincare section, which fills densely with points rather than tracing out smooth curves.">

.. only:: html

  .. image:: /api/gallery/chaos/billiards/images/thumb/sphx_glr_plot_sinai_billiard_thumb.gif
    :alt:

  :doc:`/api/gallery/chaos/billiards/plot_sinai_billiard`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Sinai Billiard (Chaotic)</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A billiard particle moves in a straight line at constant speed inside a closed boundary and undergoes specular reflection, \mathbf{v}&#x27; = \mathbf{v} - 2(\mathbf{v}\cdot\mathbf{n})\,\mathbf{n}, whenever it strikes the boundary. This billiard&#x27;s boundary keeps the major arc of a circle of radius r, x^2+y^2=r^2, and closes it off with a straight chord at x = r - c, where c (``cut``) is how far the chord is cut in from the circle&#x27;s edge. Slicing this flat chord off a circular billiard breaks integrability without making the system fully chaotic: depending on how large c is, the Poincare section shows a mix of regular invariant curves (surviving islands near the untouched part of the circle) and a chaotic sea near the chord. This &quot;mixed&quot; behavior sits between the fully integrable Circle/Rectangle billiards and the fully chaotic Sinai/Stadium billiards.">

.. only:: html

  .. image:: /api/gallery/chaos/billiards/images/thumb/sphx_glr_plot_truncated_circle_billiard_thumb.gif
    :alt:

  :doc:`/api/gallery/chaos/billiards/plot_truncated_circle_billiard`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Truncated Circle Billiard (Mixed Phase Space)</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/chaos/billiards/plot_bunimovich_stadium
   /api/gallery/chaos/billiards/plot_circle_billiard
   /api/gallery/chaos/billiards/plot_ellipse_billiard
   /api/gallery/chaos/billiards/plot_extreme_sensitivity
   /api/gallery/chaos/billiards/plot_poincare_comparison
   /api/gallery/chaos/billiards/plot_rectangle_billiard
   /api/gallery/chaos/billiards/plot_sinai_billiard
   /api/gallery/chaos/billiards/plot_truncated_circle_billiard