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):

enter image description here

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):

enter image description here

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:

enter image description here

And, unsurprisingly, it's very similar, albeit "rougher" and less symmetrical. Here are the iterations I've done (total about 5 hours of calculations):

enter image description here

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.

  • 1
    $\begingroup$ You could plot any plane through your four dimensional parameter space. You can also get three dimensions by creating a animation. Mathematica can allow you to manipulate the parameters to find interesting results. If you would show your code I'm sure we could make it fast enough to make these suggestions practical. $\endgroup$
    – wxffles
    Dec 1 '14 at 7:49
  • $\begingroup$ Yes, please add code. Otherwise this will be too broad and answers will be based on speculation and not helpful for other users. $\endgroup$
    – Yves Klett
    Dec 1 '14 at 7:52
  • $\begingroup$ There's so much code that it might not be worthwhile. These patterns are taking hours to generate. $\endgroup$ Dec 1 '14 at 9:09
  • $\begingroup$ For a 2D plot maybe use only theta3 and theta4, which should have the most chaotic behavior? $\endgroup$
    – shrx
    Dec 1 '14 at 12:19
  • $\begingroup$ Perhaps I should have stated which I've already tried. Sorry guys, my question is really broad and almost useless. I set theta1=theta2=Pi/2 and then varied theta3 and theta4, and didn't notice anything meaningful. Also, to give you an idea as to how long these take: the last picture took 30 minutes to generate, the "next one" (which makes each pixel about 25% smaller--not even half!) took about an hour, and I'm working on one right now which goes to about 75% smaller. It's been going for 2 hours and I don't think it will stop anytime soon. $\endgroup$ Dec 1 '14 at 12:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.