3D Dome Cavity SOS
Fractally Patterned Poincare SOS
This is a surface of section that gives a phase space portrait of the ray dynamics of a nearly integrable truncated dome cavity . The x-axis the the azimuthal ray coordinate from theta=0 to theat=pi/2, the y axis represents the momentum of the ray in the theta direction (or the direction it was heading when it made a bounce). This SOS is generated by taking a look at rays with a large starting azimulthal momentum component.
So far, it looks like every point is a circle (part of a stable repeating trajectory). However, the smaller cirlces get, the more phase space they span (i.e., they jump around more.
(link to higher resolution version of identical image).
(Link to 8x zoom of small image region) Many points that initially look like they may be part of the chaotic sea are actually extremely small stable island chains. As these islands get smaller, they tend to jump around more on a rouch radius defined by the spokes surrounding the central stable island.
Above is the folded region on the RHS of the SOS, that represents bounces off of the planar portion of the cavity.
When the initial conditions are set with finite starting ptheta, different regions of phase space are excluded in the trajectories, and the fractal patterns change in character.
As the deformation is increased, the chaotic nature of phase space becomes apparent. This cavity has 5 times the more of a departure from a perfect integrable hemisphere than in the previous examples.