MOTION IN PHASE SPACE
Homework
Several of you expressed some confusion about the idea of phase space as described by Gleick. Phase space makes the most sense when you are dealing with a system that has more than one dimension. Still, I will try to draw from what you already know about the logistic model to help you understand the most important features of a phase space picture.
Showing how the system behaves over time is one method of display, while the web diagram communicates the same information with a different method of display. That is all you are seeing at the bottom of page 50: two different methods of displaying information. His examples are not from the logistic model, however, and the axes represent different quantities.
As I see it, the time series graph a very literal way to show information. As you read the graph from left to right, you see exactly what happened to the sequence: it bounced back and forth as it got closer and closer to a single number. The web diagram shows this too, but you have to learn how to read it. It's most helpful to see a web diagram in motion, because then you are seeing the progress of time as it happens. Looking at a "finished" web diagram isn't as helpful, especially when you're first learning how to read them.
The same is true for Gleick's pendulum pictures on page 50. Those things with all the loops are "finished" diagrams, so you didn't get to see them emerge over a period of time. Luckily, there are some Java programs available on the internet that will allow you to see a phase space portrait in progress. Go to this link:
http://www.physics.orst.edu/~rubin/nacphy/JAVA_pend/COMP/
Some of you are really going to love playing with this program. It shows you the same information with three different methods of display. First you see the pendulum itself, oscillating back and forth. At the bottom of the screen, you see a time series graph. Then above that is the portrait in phase space. Everything is displayed simultaneously, as it occurs over time. The only slightly confusing thing is that the program constantly readjusts the size of its display window for the graphs, so you have to get used to that.
Once an experiment is done, you will see a completed time series and a completed phase space portrait. You can let it run for more iterations if you wish, by changing the "duration" setting on the control panel. Unfortunately, the most it will run for is 100 iterations. Then you have to hit start to get it to run for 100 more. I'd like to see a feature that allows it to run indefinitely.
As you can see, the axes for phase space are labeled as "angle" and "velocity." (Gleick confused us on page 50 by saying "speed" instead of "velocity.") Keep your eye on the picture of the pendulum and you can see that counterclockwise motion records as positive velocity, while clockwise motion is negative. Speed can't be negative, but velocity can because it incorporates both speed and direction. Gleick actually explains this on page 136, with some helpful pictures.
The angle of rotation is being measured in radians (if that's familiar to you). A full rotation counts as 2p radians; half a turn is p radians. The angle will stay between -p and p unless the pendulum gets all the way up to the top and starts running in multiple revolutions, like a ferris wheel. In that case, each additional revolution will be counted as an additional 2p radians, so you may start to see some crazy numbers for angles.
Each point in phase space represents a possible "state" of the pendulum. It could be sitting at angle zero, with no velocity at all, but it could also be passing through position zero at a high rate of speed on its way to somewhere else. Each of those is a different state, and shows up at a different location in the phase space. Like the logistic model, the pendulum model is deterministic. If you tell me the parameter settings you've chosen and the current angle and velocity of the pendulum, I can be quite certain where it will go next. Its path is determined by those values, which are fed into equations.
The control panel shows several different parameters which will affect the pendulum's trajectory. To see a very simple case, set the damping constant to zero. Damping is the air resistance and friction that makes the pendulum eventually come to a stop. If you set damping to zero, you have an ideal situation where the pendulum will never stop. Set the driving force parameter to zero also. Driving force is like a parent pushing a child on a swing. Without any external force to give the pendulum a boost or damping to slow the pendulum down, it will go back and forth in a very regular fashion. You can see this regularity displayed on the time series graph as a sinusoidal wave. In phase space, it shows up as a simple loop. Even as time progresses, the phase space picture will trace over the same loop over and over again, showing that it is repeating all of the same combinations of angle and velocity. This is similar to a web diagram caught in a cycle. It, too, retraces the same steps over and over.
Keep the driving force at zero, but change the damping force to .08. This means that the pendulum will run out of steam and eventually come to a stop. Watch how that plays out in the graphs. For this setting of the parameter values, you will see convergence to an equilibrium at 0,0. Remember those are phase space coordinates, so it actually signifies angle of 0 and velocity of 0--namely the resting state of the pendulum. The corresponding time series graph is known as "damped oscillation." It looks like a sine wave, but the peaks get smaller as time goes on.
Once you feel like you know what it is that you're looking at, play around with all of the various settings. See what types of phase space pictures you can create. If you're lucky, you might find a way to create a "strange attractor," as described in the book.