Help explore a self-feedback camera-monitor chaotic system We are trying to emulate the chaotic system Jim Al-Khalili demonstrate (3 min video).
In our chaos lab, we are trying to research the chaotic system shown in the video.
We are using just a webcam and a regular PC monitor. Our goal is to build a bifurcation tree for this system and to show how by changing the parameters of the system (location and distance of the camera from the monitor, time delay (between capturing and showing what is captured on the screen)) we can see a transition from "order" to chaos.
The problem is we are not sure what exactly is the mathematical representation of the system, and what is the thing that is "doubling" and going to chaos (the y axis of a bifurcation tree - like voltage peaks on the diode in a chaotic RLD system).
How do we approach this subject (if it's even possible)?
 A: I left a comment saying I would get back to you about Jim Crutchfield's work in this field, but then forgot to do so. You can find several relevant papers by doing a Google Scholar search for "Crutchfield video feedback", but I think the main references are
J. P. Crutchfield (1984) Space-Time Dynamics in Video Feedback. Physica D. PDF link.
and
J. P. Crutchfield (1988) Spatio-Temporal Complexity in Nonlinear Image Processing. IEEE Transactions on Circuits and Systems 35(7). PDF link
In these papers you will find mathematical models of these types of phenomena, including the application of bifuraction theory.
You may also find a colourful video on the subject by Crutchfield on YouTube, which has lots of examples of the dynamics that can arise in these systems. 
A: The webcam + monitor loop demonstrates a recursive "Droste effect"
http://en.wikipedia.org/wiki/Droste_effect#mediaviewer/File:Droste.jpg
which the Wikipedia article describes as a visual example of a "strange loop", a self-referential system of geometry instancing. The resulting still image has self-similarity http://en.wikipedia.org/wiki/Self_similarity (i.e. the whole has the same shape as one or more of the parts) which is a typical property of fractals. When the foreground object moves, we see its image processed recursively through a 2-D discrete-time filter that, due to feedback, might have infinite impulse response if video level were not limited. Frame delay multiplied by a large number of visible image generations means the system takes many seconds to stabilise after an object movement.
I do not see an obvious way to demonstrate a transition to chaos instructively here. I have seen interesting oscillations in the same setup where the video camera has a fast automatic gain control that hunts for the average brightness.
A: The y-axis on the bifurcation diagram tells you the fixed points (and sometimes limit cycle extrema) of a variable as a function of the bifurcation parameter.  
For example, a sideways parabola demonstrates two fixed points moving farther away from each other as you go along the bifurcation parameter.
In the case of the bifuraction tree, it's often the state mapping that is being plotted, every time you see a new branch point, it represents a period emerging in the system.  It is often called "periodic doubling", as it is the periods that doubles.
