The Duffing equation gets a little bit more interesting. So, I'm just showing here v is equal to a unit a and a unit b, so I just have, basically, x dot is equal to x double dot plus x plus x cubed. That's what I'm simulating in this one. But, this gives me now that phase space plot we're talking about with the Duffing equation, and there's a reversal of signs as well. This part would be stabilizing; this part is actually destabilizing. That's kind of what we had with the omega two motion. That was that case, and this is the phase space. If these have opposite signs, your x double dot can go to zero if all the x's are zero, which is kind of makes sense. That's right here, and around it we get something that looks a lot like an oscillator equations, nearly circular stuff. But, you can also get an equilibrium if x is, you know, if x is equal to one, then you have minus one plus one cubed just still one gives you equal to zero. So, plus one is one and minus one. So, those are two equilibriums that we can read out from this phase space plot, that we can do. Then, we can have some fun. We can play with this and make this a little bit bigger. So here, I'm plotting the same phase space plot of all the possible, you know, lots of initial conditions. If you started up here, this is what would happen to your curves and I'm plotting for a fixed amount of time. This is a way then I can show you, numerically, the separatrix motions. We saw it in the video, those, that spinning handle, right? It hung out there for a certain amount of time, and then it went back to the other side, and came back. And, we talked about how you hang out near the separatrix motion, as you get there. So, here, what I've got something is I can put an initial condition and it quickly solves the differential equations, and shows me the resulting phase space motion. If I put my initial condition up here, you know, starts very big. I had a cube, an unstable cubic term, so that one would go off to infinity, as expected. Right? If I'm really, if I have small departures, stable linear term keeps me stable but if I get too far, at some point the unstable cubic will dominate. And that's what's happening. If I reduce my integration time, you will see that curve gets smaller. So, you can see now what happens. Here, I mean, the linear part, I'm just doing a fraction of an orbit. But, as I start to get close to the separatrix motion, you can see I don't get very far. I'd have to get really precise. So, as I increase, I'm not exactly on it. But you can see as I'm increasing, that curve gets close and really hangs out there for a while, and eventually it moves back. That's precisely what we saw happening with that handle that the astronauts spun, right? It hung out there for a while. If I could get more precisely on it, I mean it is very very sensitive. If I'm off, it hangs out there even longer. And if I were perfectly on it, it would hang out there forever, and never quite get there. If you're off infinitesimally, and you go in a whole different place, right? And that's the definition of chaos, infinite sensitivity to initial conditions. You're smidgen to left, and you go off there. You smidgen right, you go somewhere completely different. That's why this is just a, hopefully, visual illustration of why we're talking about separatrix motions and the conditions. So, it's kind of fun to play with. But as we come in and have enough time to actually complete my phase space plots. I'm also showing speed in here somewhat, so as I'm coming in. I don't need that. Anyway. What was I talking about? Speeds. Yes, this is showing something about the speeds. Here you got green is faster, blues and purples is very fast, but once you come in, past the greens, yellows, so the reds that gets much slower again. If you're close to the equilibrium, you just have a nice slow, just a little bit of a wobble. And if you get close here you really slow down. But then around it, we speed up to flip back again. So this is just kind of a nice way to close out the torque-free motion part for a single rigid body. We've seen the video that illustrated it. We've seen up whole hold plots that do it graphically. We've seen mathematical formulations where we did the axisymmetric and general cases, and the different predictions that you can do about these different motions and how things behave.
