This will seem like a physics question, but I'm looking for something to do in Mathematica specifically. I've successfully modeled a quadruple linked pendulum by setting up the ODEs and solving them using
NDSolve. I've done a few interesting things with the simulation. I've shown angles vs. time, and how a tiny change in initial conditions result in a completely different, unpredictable path for the masses of the pendulum. I've shown how the time for the farthest mass to "completely change path" itself changes with initial condition. I've shown how the "time for the farthest mass to flip" changes with initial condition. Now I'm plotting something like this (this is for the double pendulum):
Wikipedia explains this as follows: "Graph of the time for the [double] pendulum to flip over as a function of initial conditions" (see https://en.wikipedia.org/wiki/Double_pendulum#Chaotic_motion for more details). Now I've replicated this quite well (this is for the double pendulum):
I'm looking for similar patterns in the quadruple pendulum. The biggest caveat is that there are now four initial angles to change, and plots above only have 2 dimensions. So I've been playing around with which of the two to vary, and which two to leave constant. I've also tried setting all four initial angles to vary with only two angles (like theta1 = theta3 and theta2=theta4). I haven't been able to get anything interesting to show up, except in the case where theta1=theta2 and theta3=theta4...which is basically like a double pendulum:
And, unsurprisingly, it's very similar, albeit "rougher" and less symmetrical. Here are the iterations I've done (total about 5 hours of calculations):
So my question is: if I can only plot two angles (as above), but I have four initial angles to play with, is there any combination I should adjust that might lead to an interesting pattern? I ask because generating these plots takes a long time, so I can't just try a bunch and look for a nice one. More generally, are there any other "interesting" questions that I can talk about now that I have the simulated model (I have things like the momentum, angles all as a function of time)? Specifically, things that Mathematica could do now that I have the solutions to the angles as a function of time.