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over here since I have some extra space. Now, I mentioned P equals 3 and Q equals 2 as an interesting case. It's interesting because something that we have not seen before can happen. All our closed orbits in the course so far have lived in the plane or on the cylinder. But now that we're
 on the torus, you could kind of think of the torus as living in a three-dimensional space, like a real donut does, or an inner tube of a tire. And if you take this case, P equals 3 and Q equals 2, the closed orbit visualized in three-dimensional space will have a knot tied in it. So you may have wondered, can limit cycles have knots? Yes, they can. Here's an example. These aren't
 These are not isolated closed orbits, but still you can have closed orbits with knots in them. This gives... The closed orbits in this case are what in knot theory are called trefoil knots. So a trefoil knot is a
 The coil knot is the simplest knot that's similar to the knot you would make in your shoelaces if you then glued the tips of your shoelaces together. So it looks like this. This is how it's often drawn up. Some people call it a clover leaf knot. Let's see here. So it's a little tricky to do these knots.
 I'm going to go over here. And this is under. That's over there. So that's a trefoil knot. Now what does that have to do with our torus? Maybe you can see it. That is, draw the torus from the top view. There's a hole in the middle of the torus.
 So imagine what it would have to look like if we go three times around one of the circles while going two around the other. You can see that there's a trajectory that goes sort of like this. I'm on the top of the torus now. Then I dive into the hole. Now, when I'm in on the bottom of the torus, I'll continue to rotate and I come out over
 Now I'm visible again on the edge of the torus and now on the top. I'll come around, go into the hole again. Now I'm underneath and continuing to rotate I gotta go something like... No promise that this is going to work out right. I'm underneath.
 resurface on the top, going this way. And what else? I have to dive one more time into the hole, once more into the breech, underneath, coming around, and then reconnect over there. Well, that's a pretty lopsided looking trefoil. Can you see that I've tried to draw, well, it's a little bit
 This thing's... that's the beauty in topology, it's allowed to be tilted. So I'm off, kind of. But you get the idea. Okay, if I want to draw it a little more accurately, maybe it should be a little more symmetrical looking. This would have to be about like so, and then...
 Something like that. That's not too great either, but okay. Anyway, what was my point? That I've got this closed curve that's got a knot tied in it in three dimensions. Okay, maybe you don't care about that. But I think that's pretty cool. And this will occur whenever P and Q are relatively prime. It's not anything special about 3 and 2. So these are called torus knots.
 for obvious reasons when you have p and q relatively prime. Is there any question about what I just showed there? OK, so that can occur even. Yes? They're not all necessarily triplets.
 No, they're not trefoils. What I'm trying to say is that the generalization when they're P and Q is then they're called PQ torus knots. So the trefoil is the special case 3 to 2. But, you know, if you had 5 to 3 or 11 to 8 or whatever, those are all different kinds of knots. Yes? Relatively prime, no common divisors. So they're not prime numbers. Like 8 to 3, 8 is not a prime number.
 but 8 and 3 don't have any factors in common. So that will still make a knot. But if they do have factors in common, like 4 to 2, that's not going to be a knot, I don't think. Yes? Well, I also don't want to
 to do a case like 5 to 1. So I want p and q, so I'm just trying to be a little bit careful. Yeah, so something like 5 to 1 just looks like a spring, you know. But yeah, you're right, otherwise you can make them relatively prime. So this is a very common phenomenon in that sense. Okay, but on the other hand, rational numbers are themselves not a common phenomenon, right? You may have learned in some other course that if you
 If you look at real numbers, almost all real numbers are irrational. The rational ones are very exceptional. They have measure zero would be one way to say it, or they're also countable. So there's two different ways in which the rationals are very small. If you haven't run into these ideas of
 uncountable, uncountable, or measure, don't worry about that. We'll talk about that a little more when we get to fractals. So having said that, let's talk about the irrational slope case because that's
 That's where something really new can happen. I mean, yes, OK, we have these periodic orbits with knots tied in them. But something really even stranger is this case 2, where suppose the omegas are not going to create a rational ratio. If omega 1 over omega 2 is irrational,
 Like say omega 1 is the square root of 2 and omega 2 is 1 or something like that. Then we get quasi-periodicity. And so what that would mean is, I can't draw it really, but I'll just try to indicate it.
 This is the case where you're going up, but you don't ever close. You just keep going. And so if you just keep drawing this picture over and over, the lines will never touch each other. They'll just keep going and going and going, and they never
 close, which might sound to you, okay, so there's a tendency in physics to say it a certain way, which I don't like, which is that sometimes they'll say that a quasi-periodic trajectory goes through every point on the torus. That's not actually true. It does not do that. It doesn't go through
 through every point on the torus, but it is dense on the torus. So that's the official way to say it in math. That is, every trajectory on the torus does not close
 So how do we know it doesn't close? If it closed, that would mean that we could count the revolutions. It would have gone some integer number of revolutions around each circle, and you'd be back in the p over q case. So it can't close. So given that, and every trajectory does not close, and
 And every trajectory is dense on the torus. By that I mean that you could do this little test. Pick your favorite point on the trajectory, or pick your
 What's your favorite point on the torus? Anywhere. Like this point has not currently been hit. Will that point eventually be hit by a trajectory? No, not necessarily. However, if I draw an epsilon disk around that point, where epsilon can be as small as I want, as long as it's not zero, the claim is that the trajectory will penetrate that epsilon disk eventually.
 Okay, that's the statement. That the trajectory gets arbitrarily close to every point, but it doesn't necessarily hit every, it doesn't hit every point. It gets epsilon close to every point where epsilon is as small as you want. So that's the statement. So in that loose sense, you could say the trajectory goes everywhere. It certainly comes close to everywhere, but it doesn't literally go everywhere.
 So it's kind of amazing that this thing can just keep going. I mean, why is it amazing? To me it's amazing that a trajectory can live on a compact set like a torus, a closed set, and keep going forever and ever and yet never cross itself. It doesn't seem like there's enough room for that, but there is. And that's what it does. Now you could ask, is that chaos?
 It's not chaos. It's a milder – it's still weird, but chaos is weirder than this. I'll explain why later, but you have a question. How do I know it doesn't close, or how do I know it's dense? Yeah, how do I know it's dense? I might want to give you a homework question about that. I'm not sure I want to say,
 that's something about that. I think if you try to think about it, it's a good exercise. Yeah. So let me not give you any ideas about that. Was there another question? Okay. So I said, of course, I haven't said what chaos is yet, so it's meaningless for me to say this is not chaos. But the key thing
 I'll just give you a little more shadowing of what we're going to discuss. Suppose you compare two neighboring points, like say this point and one on a trajectory near it. Think of them as sort of like buddies and they're running along on the torus. You want to ask how far apart are they as time goes on? Do they diverge from
 from each other exponentially fast. In a chaotic system, they would. That is, nearby states get exponentially separated, at least at short times. They can't exponentially separate forever and live on a closed, you know, bounded space like a torus because they can't get farther apart than the diameter of the torus. But when they're close initially, in a chaotic system, there's exponentially
 whereas here you could check just by exactly calculating theta 1 and theta 2 of t. They're linear in time. So the distance between these two, while it will grow, it will grow linearly in time, a proc. It does not grow exponentially fast. So nearby trajectories don't separate very fast in a quasi-periodic system. And for that reason, they're not consistent.
 where do these come up? They come up all over the place in classical mechanics. So it's very common to see them. I guess I just wanted to say one thing about the coupled oscillator case, and then we'll stop. So back to the coupled case. Let's look at that for a second. I have the equations up
 there on the top board. And so if I introduce a variable phi to try to understand what's going to happen in my coupled oscillator case, it's natural to look at theta 1 minus theta 2 because our equations, which remember looked like this, theta 2 dot is
 omega 2 plus k sine theta 1 minus theta 2. Both equations in the sine function, they only depend on the difference of the angles. And what this means is, like in terms of the runners, that there's coupling between them in the sense that if theta 2 is a little bit ahead of theta 1 on the circle, and let's think of k as
 being a positive number, then if this is a small angle difference, this will be sine of a small positive number which will be positive, times k which will be positive. So what's going to happen is we're adding a positive number to omega 1, and so it means in a situation like this where they're both running this way, the runner who's behind will get a positive term added to his or her speed
 meaning the person behind will speed up and the person ahead will slow down by this amount. So it's a coupling that tends to synchronize the oscillators, the runners. It tends to pull them together in phase. And that's something you would want if you're running with your friend who's a slowpoke. You want that person to speed up, come on, catch up with me, and you might slow yourself down so that you can stay together and talk. Anyways, that's what this
 because model is about, it arises in other settings, but let's not discuss that now. Anyway, so with this phi, we're looking at this angle between them. Phi would represent that angle. And by subtracting the equations, you can see phi dot would be, well, let's see. So it's omega one minus omega two. And then we have, it looks like minus two K,
 and you might recognize that because that's the equation with different letters that we discussed earlier when we were talking about the infinite period bifurcation, the sniper. So the same analysis will apply and what we can see is that when omega 1 minus omega 2 in magnitude is less than 2k,
 Then we'll have two fixed points on the... V-star will have two fixed points. Sorry, there's two fixed point solutions. What am I trying to say? Two fixed points for V-star.
 diagram on the torus would look like this. Lines where phi is constant, in this picture, phi equals constant would mean theta 1 minus theta 2 is a constant.
 and that will mean that we're moving on a diagonal line. So we would have a picture like this. Diagonal line with a slope of 1. And then we'll also have another diagonal line which is going to turn out to be corresponding
 to the unstable phi star. That is when I say there's two of these fixed points. One will be stable, one will be unstable. And you can check that they're going to look about like this. And that the flow takes you onto this curve away from these curves.
 So, what I'm saying is that in this system where the K is sufficiently large in the sense that I just wrote down to K greater than the difference in the frequencies, that the system will end up running on this kind of a diagonal line which corresponds to what's called
 that is the phases will be locked together with V1 minus V2 equals some stable distance. That is the runners will be synchronized. They're not going to run side by side, but they will keep a fixed distance between them asymptotically. And if BOMAT is not in this regime, then they won't be able to run.
 together one will continually lap the other. All right, thank you. I think I'll stop there.
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