A lesson about interactivity. Having realized that I should be using quiver plots to understand the dynamics of 2-variable differential systems, I went back to the Mexican wave and made a quiver plot. In Matlab it is a slow task to tweak a parameter and re-plot over and over again so I made a zlab (C++) version of it that allowed me to twiddle the parameters in real-time.
What a difference! What took me a week before trying to understand the parameter space of this system now took me only 3 minutes. It is proof again of the power of interactive twiddling. When twiddling the variables the fast response and ability to "scratch" back and forth allows you to quickly intuit both the action and derivative of each parameter. When twiddling I was saying things like: "Oh... this is causing all the arrows on the left to go up" and "this is moving the steady-state point." I understand this system a lot better now.
In the following plots, I show two time traces at different places in space as indicated on the northwest space-time plot.
Plot 1: the stable Mexican wave. The third quadrant of the phase plot means "sitting & excitable (not tired)". Note that there is a stable equilibrium in that quadrant so if the system ends up anywhere in the 3rd quadrant then it falls into that basin and stays there until disrupted by, for example, a neighbor who pulls it towards the standing side.
Plot 2: two stable points. In this configuration, the 3rd and 4th quadrants have stable points so it simply transitions from the starting point in q3 until it gets knocked into q4. I guess if I were to hit it with a pulse of "tired" then I could get it to transition back again so this system is akin to a 1 bit memory. Haven't decided how to take advantage of this yet, but I'm sure there's something cool to be done with it.
Plot 3: oscillator. Now the equilibirum point has been removed from Q3 and so the system just perpetually oscialltes. Note that the oscillating attractor is stable -- the green and blue traces converge into the same limit cycle. I'm not positive what gives it this property, I thought that diffusion might be helping to stabalize it, but that's not the case as shown...
... in the following with no diffusion.
Plot 4: a system that has the Q4 equilibrium right on the boundary with Q1 so that it momentarily seems to be commit but then gets sucked into a small oscillator around the origin. This oscillator, however, does not appear to have a stable limit cycle so it dampens out.
Plot 5: a pattern forming system. Here Q1 and Q2 have a huge shearing force compared to Q3 & Q4. Somehow this seems to make it into a chaotic attractor but I don't really understand how yet.
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