Diagram 10: Phase diagram / Bunimovich stadium
This time i'll show you how to read phase space diagrams. But first, some billiards:
Boltzmann famously invented statistical mechanics by thinking of atoms as billiard balls. Artin examined billiards in the hyperbolic plane. Later, Birkhoff considered smooth billiards - the ones whose border is (infinitely often) differentiable. He then investigated which of those might be integrable . That is supposed to mean that a path can be predicted because it is periodic or nearly periodic. - a case studied before by Poincare, who was trying to describe all billiard paths as nearly periodic. Birkhoff was able to show that the elliptic table is indeed integrable, and conjectured that all integrable smooth billiards are elliptic:
G. D. Birkhoff: "On the periodic motions of dynamical systems" , available here:
It turned out that his definition of integrable wasn't precise enough, and others had to improve later on to make the question well-posed. So today, there are multiple answers and some questions still without.
Before we continue, here some specimen: Integrable billiards : Rectangle, circle, annulus, ellipse, ... non-integrable billiards : Sinai billiard (a square with a circular hole in the center), cardioid, non-concentric circles, the stadium, ... .
Now let's take on the the hard, chaotic part.
In this post's picture you can see a Bunimovich stadium billiard. For those, most paths are ergodic (that is get close to any point in phase space). You might have guessed that the physicist's "phase space" is related to what i'm getting at, and you'd be right. Luckily the phase space for a discrete billiard can be made 2-dimensional, and here's how:
At each reflection, we have a position where the ball hits the boundary, and a corresponding incident angle . We're not interested in the parts of the trajectory where the ball doesn't interact with the walls. So let's record angle and position only when it does hit the wall. That's called a Poincare section, though, i have usually seen this term for systems where the path outside the section is continuous, and also interesting. Such as to reduce a continuous problem to a discrete one, but i guess i was just being narrow-minded.
The pair of plots in the middle are the phase space diagrams. Each color represents a specific trajectory and any point a piece of reflection data. Interestingly, the points of a color often lie on certain curves. So as long as the phase space diagram isn't completely full of noise, we can see much of change of the behaviour implied by varying parameters. Remember that one can only show some orbits, since otherwise there'd be a data point everywhere.
I got the pictures from this page, but you'll also find movies and more explanations here:
Nice quick gif animations "mixing in diamond billiards":
If you look for a simpler example, maybe start here: "Billiards in Nearly Isosceles Triangles" W. Patrick Hooper and Richard Evan Schwartz
bits & pieces:
You can reflect a table along stright lines (or fold it along a symmetry axis)
All paths are reversible. That means that the chaotic parts of the mushroom will enter the pit again.
The Bunimovich mushroom is kind of both: integrable and non-integrable. Along the hat we have all circular paths with angle small enough not to hit the pit. In the pit we'll have some periodics but most are ergodic.
For real chaos we need stretching and folding. Focal points often lead to dispersing billiards . But Watch out! Lemons the stretching in lemons is counteracted so that only limited folding seems to occur or so. I'd love to know more about these.
Those wavy pictures, at the bottom. are done using cellular automata (someone should have told me years ago)! Straight lanes without waves are calles "scars". Those are good places to look for peridoc paths. The word eigenfunction has nothing to do with the phase space diagrams. It means that after iterating the cellular automaton for a while the picture won't change qualitatively anymore.