 Good morning. As Mateo said, my name is Ramaswamy and I'm presently at IIT Delhi. In this course of lectures on the collective dynamics or collective behavior of some kind in complex systems, I want to largely discuss the dynamics of couple systems, and for reasons that are almost obvious. One looks at oscillatory phenomena. There is not just transient behavior that might go away after a while or not just things that never change. So we want something that changes in some fashion. Most of the time it will be periodic motion, that is motion that repeats exactly after some specified amount of time. Or it could be quasi-periodic, which is a little variation of periodic behavior. Or even chaotic behavior, because that is somehow the most common behavior in the world, right? Now the kinds of things we will talk about, many of you may already be familiar with it because synchrony especially with Steve Strogatz's book and podcasts and infinite number of TikTok videos and so on and so forth. This is a very common thing, but I'll also talk about its variations. I want to talk about things that have names like death, amplitude death, aging, stasis, etc. And also more quite interestingly symmetry breaking, which is a dynamical symmetry breaking which brings all this into the realm of statistical physics, there are phase transitions, and so on and so forth, okay? So this roughly is the plan. The topics that I will try to cover is what is synchronization? Starting with the oldest observation, at least the oldest scientific observation by Christian Huygens. I'll talk about that experiment, its theory, and how we understand it. I'll also try to then segue into a discussion on phase transitions and synchrony, what is the similarity between the two? And this is particularly important in a model which can be solved exactly, it's called the Kuramoto system, the Kuramoto model. Because chaotic synchronization is in a sense one of the more interesting aspects of synchrony, how do systems that never repeat themselves and are unstable everywhere and so on and so forth. How do those systems synchronize? Now that is also an important aspect of the subject. And then there are variations called generalized synchronization, etc, etc. But I will try to talk about it from a geometric point of view, using the ideas of manifolds and so on. Then as I mentioned already, symmetry breaking is going to be important. So I will discuss what are called chimeras. Then I'll talk about networks because nowadays, all complex systems are networks, etc, etc, etc, etc. And finally, one of the more interesting recent-ish developments. The subject's going back to 1665, so from 1665 to today, there's been a lot of developments, particularly after the discovery of chaos in this area. So I'll talk about interesting thing also introduced by Steve Strogatz, called Swarmilators, namely things that swarm and sync, okay? So this is something of current interest. Now, so one of the reasons one studies synchronization is that this is one of the most common forms of collective behavior that's seen in nature. I mean, there are simple examples that surround us everywhere, particularly from biological systems. All of us are examples of something that is very heavily in sync with the sun because we depend on the sun for so much of our energy, that it's always useful to have processes that keep some kind of a 24 hour cycle, because that's where the earth's rotational period is that. So such rhythms are called circadian rhythms, and these are there in virtually all organisms from bacteria onwards. There's a lot of synchronization that's going on in our brains. A lot of neurons spike simultaneously in order to retain some forms of memory. Which ones are spiking together? So there will be, there is a correlation between the kind of memories you form and the kind of synchrony that sounds or various external stimuli induce in your brain. If possible, I will talk about some current examples that I've studied in neuroscience. Other social phenomena which are known to be synchronous is, for example, when groups of people walk together, then they fall into patterns. Particularly if they know each other well, like groups of friends of varying heights, so varying lengths of steps, you might have noticed that when people go all together, they start more or less walking together and so on. So this is a social phenomenon which is an example of synchrony. That is, that is in different units. And one of the things that's necessary is some interaction, right? Okay, before I put on the video, let me just show you another thing that happens. If you go to a stadium, you find Mexican waves forming, right? And one person starts and then suddenly it goes on. Or if you go to particularly good performance, you find synchronous clapping, people get into a mood and then that, that, that, that, you know, it goes on, right? This crowd is big enough and we could try for an example of that later on, okay? You can hear whatever sound this is. This is the Millennium Bridge in London. And when it opened on 10th of June, 2000, you can see the crowds that are, there are many videos on YouTube. So if this one is not clear enough, notice how suddenly you find groups of people who are walking together in sync. And this was bad enough that the bridge became known as the wobbly. Because it was shaking from side to side. That's particularly obvious now, right? So, this can actually be a major problem. Armies when they walk across bridges are told to not walk in step. But, you know, armies that march are always in step and they are told to stop being in, you know, break file, the command or something. Because otherwise you'll just get your natural rhythm of your walking, tuned with the bridge and bridges have been known to collapse because of that. Anyway, so they had to go and fix this. This was a big disappointment for the city of London, and a huge opportunity for people to study synchrony and chaos and so on. A number of papers that have come out on this side, quite a few. Okay, but in nature, there are these fireflies in Malaysia, particularly. The fireflies are not moving themselves, or one or two of them might be. But otherwise, they are largely static. And they flash in unison because they are trying to attract mates. Only male fireflies flash. But the synchrony that emerges, you know, just go again, right? So you can see that, this is not the best example I've seen. That I've seen, you know, entire trees just light up after a while. Because all of them are flashing together in unison. Now, they are communicating with each other mostly in response to another male's flashing. This one gets into sync and tries to be brighter or something. I don't know, more attractive in some way. So, I mean, the precise reasons, etc., are only hypothesized because these are not easy experiments to do, but you can observe them. Also, the observation of this has inspired a lot of theoretical work, trying to come up with models that could start synchronization. And this eventually led to the Kuramoto model, which I will talk about in some detail. But, and here I have to actually thank our director Mateo. There are other forms of behavior that could be seen as some kind of synchronization. So, here's an example of fish that are schooling. You know, when they are not in the schooling formation, they are just moving around randomly. But something happens, maybe a shark somewhere, and then they all start forming these shoals that are all traveling together. And one can see that there is, there is order, there is short range order. So, it's random motion that is just, and because the fish are sort of long. You can think of nematic ordering if you like, if you like the language of liquid crystals or something, all right? But this definitely, you know, one can see more than just the fact that they form a group that moves together. They're all aligned in, you know, at least locally they're all aligned, right? In the same direction, because otherwise they'd be crashing into one another. So, the emergence of this kind of spatial and temporal pattern, if you like, can be seen within the framework of synchronization. And when I discuss generalized synchronization, I will talk about it. You remember your comment, I took it very seriously. All right, so here's another example, okay? And by now, anybody who studies complex systems has seen this and, you know, Nobel Prizes and so on and so forth. A lot of very beautiful work on active matter and so on. And here again, I want to point out that there is the emergence of at least short range order. They're not all flying together in some straight lines or even some generally. I mean, these are groups that form and reform and so on and so forth, okay? But the reason why I believe that it's useful to think of this as some kind of a new synchronization is that these patterns can be engineered. Swarms of drones, for example. As we have seen in recent applications, shall we say, drones can be made to swarm and that dynamics is not very different from this, right? Also, when you look at these kinds of murmurations they are called, right? When you look at them in some detail and with the eye of an artist, you find representation like you see on the right hand side over there, right? So again, what the artist is able to see, and I will show you photographs later on of the actual birds in flight, is that these, again there is short range order. There is definitely some kind of emergent property over here. Because the bird doesn't, I mean it infers a pattern to be maintained through years of evolution and all that, of course, but it's there, right? Now, as I said, the first mention of synchronization was by Christian Huygens in 1665 or thereabouts and Huygens as most of you, I presume, know did a lot of work in early days of mechanics. He invented the pendulum clock and so on and so forth, right? Lots of stuff, ray optics, right? Now what he observed was, I mean he observed this phenomenon and made some very explicit notes which he communicated to his father and so on and so forth, but what he basically saw was that if you hanged two pendulum clocks on a common, so here's the two pendulums. And here they were hung on a common beam. And what he observed was that no matter where you started the pendulums from swinging, after a while, they were almost, after a while he saw that they were swinging in a manner such that this pendulum reached its extreme, that is this position, whatever this symbol is, at a time, when it reached this, this particular pendulum was on this side. So both these pendulums were going to their extremes at exactly the same time. However, there are two ways in which they can do it. Both of them can reach their left extreme together or their right extreme together. But this was happening in a way such that both of them reached their outer extremes first and then their inner extreme so that they were oscillating essentially out of phase with one another. Now, what he observed by and large was this situation, and it's anti-phase synchronization or out of phase or whatever. And this would be the case of in-phase synchrony, namely that it would reach, the extremes would be reached at the same time. So just to make the point even more obvious, here are the two phases of the two oscillators and they are just displaced a little for clarity. So the two of them are exactly in phase, over here they are out of phase. And what Huygens saw was mostly this. He saw the other one also, but not for very long. And when you do the experiment again, three centuries later in the early 2000s, there was an interesting paper. I'm going to give you all these references so that you can read them on your own as well. What you find is that you started off any which way, and these are not long enough to see the thing, but you can see that both of them are on their outer limits at the same time. So this is the common thing that one sees. Again, from YouTube, and there are even more spectacular videos, I just let this play and then we'll talk about it. No, they're not touching each other, but notice that this baby here is moving. He's also in sync. All right, now what I wanted you to observe over here was, I'll just go back and play it if necessary. You saw how they were just set off randomly, right? And then they came into a stage where two of them, they were actually anti-phase. That wasn't stable, right? And then it went out of sync for a little while, and then it came back and where everything is now in phase, all right? So the pendulum hanging from the top was largely outer phase. The system, these are metronomes. So the metronome system, it's largely in phase, right? And you can, like I said, and this is only with four, but you can see thousands of them if one likes, all right? Now what we are seeing over here is demonstrations of this general idea of synchrony. And no, this is a periodic, right? Because this is, metronomes are used for musicians to keep time. So they need definite beat. So these are all periodic, right? You cannot actually make a chaotic metronome. I mean, I suppose you could, you can make anything chaotic. But metronome by itself, if I just left it on the table, is designed to be periodic. There's a spring that you wind up, this is actually not very expensive, right? And so you wind up the spring that releases energy and then it just keeps the time. Usually people go for an eight beat, if you're a musician, when you want to do an experiment, you go for a little faster stuff, yeah, very close, all right? And this is going to be the point. You know, if two identical systems, everything identical, identical, identical, perhaps one is not so surprised because the motion has, at some point at least, tell me your names so that I don't have to say him in the blue or him in the gray, all right? That you start out as close as possible, all right? But because there are small variations, even in a hanging pendulum, there's lots of technical things in making it as a good timekeeper. Huygens experiment actually made it impossible. He made these pendulum clocks that were good timekeepers to the level of something like 15 seconds in a day, all right? That was how accurate they were. But because they could synchronize with some comparable movement, they could not be used to determine longitude, all right? So that actually is a major problem and there's a beautiful book called Longitude, which you may like to read at some point in time because even though it's a historical book, but it has a lot to do with our history, how we invented basically how it was possible to go from my continent to yours, presuming that we are from different continents, of course, all right? Okay, yeah, yeah, yeah, go ahead. Is it important that they are on the same platform which starts oscillating as well or yeah, okay? That it starts oscillating is not important. That it is not infinitely heavy so that it damps out any, you know, what is necessary is for one system to recognize through the medium that the other one is there, all right? Yeah, it works. When it's very heavy, I will discuss some examples, okay? Yeah, see, the things that are important in all this is how different are the natural frequencies of these isolated systems? If they are very close to one another, then one kind of synchrony is possible. If they are not, they will, I'm just going to discuss that in a minute. There's going to be another kind of, there are other possibilities, let's say, all right? But that's important that they be close. The other thing that's important is the amount of damping in the both intrinsic damping through the medium maybe and the strength of the coupling, which is really dependent on the mass of the base or the suspension and the mass of the pendulum and so on. I mean, I'm not sure that those are, I mean, those are all clubbed into one. The theories of all this are phenomenological, right? So it's not as if you're looking at material properties and so on and so forth, okay? But it's not important that it be on a swing. I use this video mostly to show, because the motion is very visible, right? You can see that, but it shouldn't give you the impression that that is the necessary condition. Yeah, so now two uncoupled pendulums with slightly different frequencies are, I mean, they're going to be uncorrelated, right? So if one pendulum has frequency omega 1 and the other frequency is omega 2. If they are almost similar pendulums, then let's say if that is the condition, then coupling them somehow makes these two pendulums pull each other to a common frequency. Namely, this one will go to some new frequency omega 1, this will go to some new frequency omega 2, and those will be identical, right? So that's what we are seeing, if one was to take a measurement. The two, we saw the uncorrelated pendulums when they were starting. Each one of them had frequencies which were almost the same. But then when they were in sync, you could even hear that they were at exactly the same frequency. So this process of going from to this is called generally entrainment, okay? So the two oscillators get entrained with one another, all right? This entrainment is possible. Should I move to this side? Is it easier? No, no, no, this is not meant to be a strain on your, at least not on your body, right? So okay, see the equation of motion for a pendulum is something like let's say x1 double dot plus omega 1 squared x1 is equal to 0, or let's say i. So I mean just for small oscillations, I'm just considering it to be a simple harmonic oscillator over here, all right? It turns out that of course these two pendulum, the other one, okay, both of them are out here. Omega 1 maybe is something which is like this, where p times omega 1 is roughly equal to q times omega 2 when they are identical or close to identical p and q are both 1. Now it turns out that the process of entrainment is actually powerful enough that you can actually drag this approximate relationship of p omega 1 is equal to q omega 2 or more to the point if I've got my eyes on you, yes, integers. So this kind of entrainment is possible even when p and q are, when this relationship is there, but it works best when p by q is rational, okay? So p by q is rational and they have no common divisors, so that's down to its minimum form and this kind of entrainment works very well if p is 1 and q is 2 or vice versa, 1 and 3, 2 and 3, et cetera, you get the idea, right, these, you know, 311 by 297 is not a good idea, all right, because we'll just see why that is the case, okay? So now the solution for this is basically something like xi of t is some ai cosine omega i t plus bi, right? So this is straightforward, right? And when they are uncoupled and not bothering to couple them at all, what I would like to show you is that this dynamics essentially can be written in the form of a map, right? And we'll just see how that happens. The x1 dynamics, if I write x1 versus x1 dot is basically lying on a circle. You should stop and ask, in case this is not familiar, almost everything I say is simple enough that it can be explained in a sentence or two, okay? Okay, so this is the motion for x1 and x2, all right? So the first oscillator is going around at some frequency omega 1, this one is going around at some frequency omega 2, all right? And the combined motion therefore lies on a torus, all right? So this is mathematically S1, this is S1 and this gives us a two-dimensional torus on which the motion occurs, right? Now, the combined motion is therefore going to take place and so on. And the, is this clear to everybody that the motion of the two, no? Probably run. Yeah? Maybe for the interest of those who are in the back, if you can make, bigger pictures, okay? And in the interest of Rama, maybe next time you can sit closer. There are two options, of course, either closer or far, far away, no? Okay, I will also speak louder and hopefully little clearer and so on and so forth. Okay, now, to whom is this picked, who finds this picture not immediately easy to understand, okay? So this picture, okay, great. So we understand that a simple oscillator is just something going around in a circle, all right? Because this is just folding X and X dot onto one phase space plot, right? Now, since the two of these systems are uncoupled with one another, consider the first circle. At every point of the first circle, there is another circle. And at every point, there's another circle, circle. So finally, what you have is a doughnut, all right? Or a bicycle tube or a car tube or whatever, flora, okay? All right, so now that we have this idea, the combined motion is going to go from here, it's going around one circle. So it's going around one circle. But meanwhile, on the other freedom, which is the X2 freedom, that motion is going around on this side. So finally, the combined orbit is going around and around and around and around, yeah? Now, to try to make a map out of this, Poincaré basically said, let's just cut through this torus. And let us ask, when does the orbit come and meet this cut, okay? And now, this is going to require a little bit of faith. This is actually a two-dimensional torus drawn on two dimensions in a world which is three-dimension. Actually, this is in two dimensions and this is in two dimensions. So this is actually in four dimensions. All right? So when you cut through with this plane, you actually only get a cross-section of this torus, all right? You just get a cross-section of this torus. And we'll try to see how to see the dynamics on this cross-section, all right? And this was invented by Poincaré and it's called the Poincaré section, yeah? Okay, so to make things easy, I'm going to just remove unnecessary constants, all right? And just consider the two phases of the two systems. So when you take the two phases of the system, what you have is, let's me say, X1 is equal to cosine theta 1 and X2 is equal to cosine theta 2. I've just shifted things around so that unnecessary constants are not needed. And I want to look at the motion of the phase theta 1 versus theta 2. So, theta 1 is omega 1t and theta 2 is just equal to omega 2t, which is theta 1 divided by omega 1, yeah, all right? So now let me try to understand how theta 2 varies with theta 1. I want to get rid of 2 pi, so okay? I'm just getting rid of the 2 pi, okay? And that divides out, I just get this equation. And now theta 1 and theta 2 go, they just go, so 0 to 1 is the same as 0 to 2 pi, it's in those units. Is that okay? Just because it's unimportant. So this is theta 1 and this is theta 2. Let me just draw this line, yeah? So, but all this algebra is modulo 1. Theta 1, if it's, let's say it's 1.5, that's 1.5 times 2 pi. So I've got to remove that extra 2 pi and this is the same as this number. So essentially this is just periodic boundary conditions, right? So instead of this curve over here, what I have is, I can just fold this back over here, then I can fold this over here. And I can then fold that over there, and so on and so forth, yeah? No, okay, all right. So let me start by asking, when does it first hit this line, okay? Let me call this point psi 1. This point over here, when it hits the other integer, psi 2, psi 3 and so on and so forth, all right? So the coordinates of psi 1 are the x coordinate, sorry, the theta 1 coordinate is just going to be, let me call this alpha. Right? It's alpha, psi 1, the y coordinate is 1, the x coordinate is 1 over alpha, yeah? This point I'm going to call psi 0 and this is just 0, 0. This is the origin, right? Psi 2 is just 2 over alpha, psi 3 is just 3 over alpha, 3 in general psi n is just n times omega, where omega is equal to 1 by alpha. Sorry for a little, I want to now make the point that this is 0 to 2 pi, 2 pi to 4 pi. So this line is exactly equal to this line. Same way this line is exactly equal to this line. So if I were to just now look at this dynamics on the curve, the first point is here. This is from 0 to 1 on both sides, right? Okay, so 1 is the same as 0, so the first point is here. The second point is here. The third point is here. The fourth point is here, and so on and so forth. In other words, psi n plus 1 is just equal to psi n plus omega mod. You're not happy? No, no, no. Yeah, no, I'm just, see the whole point is that all our dynamics and phase is only interesting from 0 to 2 pi. So if I look at it in units of 2 pi, the dynamics is only interested in 0 to 1, okay? So what this number does over here is just tells you how far you go on the line from 0 to 1 in one step, again on another step. But you've got to go past 1, so that's why the mod 1. And instead of this, I can just fold it up and look at the dynamics on this is psi 0, this is psi 1, this is psi 2, and psi 3, etc. So the entire dynamics on the Poincare section is just a set. This is each time it comes back after 2 pi or 1. And so the dynamics on psi is just a rotation on the circle. How much does it move each time? One, omega, and omega is the ratio of the two frequencies. Now it's not difficult to convince yourself that the only way in which it will close is that if somewhere in the future, it has to pass through the point over here or a point over here or whatever, right? So the only way in which it will close is if the slope is rational. And why rational? Because this is now broken up into integer units, right? So if it crosses itself, if it comes back to a starting point after however many squares in this direction and however many squares in that direction, then the orbit will close on itself. Yeah, big omega has to be rational, which also means that alpha is rational, which also means that omega 1 by omega 2 is rational and so on and so forth. Yeah, I'm only interested in when it crosses a particular, see the idea of the Poincare section is basically to draw a line on this curve, on this representation and say, tell me each time the orbit crosses that line. And for convenience, I'm taking this line, right? Because this is psi 1 and the next time it's moving the same distance. Because this length, since this length is 1 and the slope over here, so this has to be exactly the same amount. I mean, this is just simple geometry. But get this because this is one of the simplest examples of what's called an ergodic system. All through Statmic we study ergodic systems, right? But this is the simplest example, well, simplest in somebody's notation. But this is a simple example of an ergodic system. If omega is not equal to p by q. Because if it is equal to p by q, then let's say it's 1 by 1 by 4, right? So as the orbit goes around this torus, if omega 1 by omega 2 is just 1 by 4, you'll get four points, nothing else. And this is very important sometimes because when you want to divide a pizza between n friends, you need to move by precisely some rational number where 1 by n, let's say, okay, yeah? Now if alpha or omega is not rational, then these points will not, after 1 by 4, it means almost 1 by 4, all right? 1 by 4 plus epsilon, which will make it irrational, okay? Then we'll go around, and by the fourth time, let's say it will actually come here, over here, over here, over here, over here, over here, etc., etc., it's too far in the lectures. Let's not talk about that right now. It's just, this is a simple thing called the rigid rotation of a circle. And so the dynamics of two uncoupled systems, all right, is going to be represented by this. Composing omega 1 by omega 2, yeah? No, if you're irrational, you will go around because you'll keep missing. You will never come back, yeah? If you're rational, you will only cover it by a finite number of lines, end of story. I mean, actually, you can see that if omega is not rational, if it is irrational, right? Then the points will eventually fill up the entire circle, okay? That is what is meant by ergodic, okay? Have you all seen this book on classical mechanics by Arnold? Seen it in a library somewhere, on your laptops? Okay, so you'll see some of it in a homework, okay? So now, we are through as far as two uncoupled systems are concerned. If I want to take this case over here, where omega 1 is approximately equal to omega 2. And let me assume that this ratio is equal to 0.9998765, where is some irrational number, all right? If I don't need to go that far, then let's say I start over here. The next time around, I will only, oops, sorry, let me just draw my circle again. On iteration 2, I'm only over there. The third iteration, I'm only over here because I'm not 1, right? Omega is not 1. So it just will keep on going, going, going, going, going, etc. And then it will not hit 0, again, it will go this way and so on and so forth, yeah? So this is what is meant by ergodic, it's just going to go around. Now, let me try to see what is the effect of coupling these two oscillators, all right? I'm going to do the mechanics of it later, but just the idea of coupling them would be to say that some function's coupling them, it doesn't matter, all right? Now, so people have been studying different forms of coupling, we'll come to that. But the simplest thing that I want to do is to actually make this map. I want to just say that what effectively it will do is to change this map. Just one small thing, let me just go back, one thing I want to just give notation. Omega is called the winding number, okay? So the winding number is the number of winds that it does, all right? So I can even, so if I take psi n plus 1 is equal to f of psi n, all right? Then I want to define it now as omega is equal to limit of n going to infinity f to the n of psi minus n. How many times I have done it? When I've got this simple map, you can see that omega, this winding number, sorry, this winding number is equal to omega for the simple map, yeah? I mean, the limit makes no sense over here, but it's going to make sense for us later on. So now I want to consider this f of n where f of n is psi n plus omega and for reasons best known to Arnold, he considered this map of a circle, okay? Mod 1 is there. Mod 1 is only important when you want to represent it, but otherwise every point is equal to itself shifted by 2 pi, any numbers of 2 pi on both sides, right? So this is a very famous map, okay? And it's called the circle map or sometimes it's just to be specific. It's called the sine circle map, all right? And when k is equal to 0 is just the rigid rotation. What happens if k is small, okay? You can see that you see psi already is a variable that mixes up these two coordinates theta 1 and theta 2, right? So what I'm writing over here is a phenomenological, algebraically convenient form of some coupling between theta 1 and theta 2, okay? And it's a nonlinear coupling. You can consider all sorts of other behaviors to be important, okay? Now, the main thing over here is that as you keep changing k, you can ask how does this winding number change, yeah? See, we were interested in the case of omega 1 approximately equal to omega 2 when uncoupled. When you couple them, do they come together or not? Yeah? Oh, it's not W index and winding number, hashtag winding number. So this is what I've just been talking about, okay? This is the, so that is it, okay? So this is what was discovered and studied in great detail in the 1980s particularly. I should point out that just one second, let me just give a disclaimer. Almost everything that I will give in these lectures is the work of many, many, many other people, all right? And I've taken pictures from the Internet with attribution for the most part. When I haven't attributed it, it's not because I don't want to, it may have just forgotten where I got the picture from, all right? This is as much for the YouTube audience if there is any, it has this. Anyway, so the point about this is that what is noticed by Jensen, Bach and Bohr and many, many other people, Kadunov, Prakashya, Jensen, I've already said Jensen, just a large number of people, all right? The following. What is shown over here is the winding number calculated from that, all right? As a function of the strength of the coupling, yeah? And what we have over here, okay? There is some interesting properties just coming from the fact that it's a sine coupling and there is a symmetry between around the line half. I'm sorry, I never got back to you on the question. F is just whatever is on the left hand side or on the right hand side over here. In our case for the map, for any map, okay? For any map which is defined like this, I will define this to be the winding number. I start at some point, I just iterate for n times, divide by the number of iterations. So I find out on average how much it has moved each time, yeah? No, no, no. In all cases, w is equal to, the first graph is just this picture at this line. k is equal to one. Just one second, just one second. Winding number. No, that's not the winding number. Yeah, I know. Winding number is equal to, this is the definition. For this particular, for the case of k equals 0, that is on this line, on this line, that was, right? Next, then I increase the value of capital omega in my map. Then I calculate the winding number, all right? So if I keep calculating the winding number, it turns out that around every rational number, oops, okay? Around every rational number, 0 by 1, 1 by 1, 1 by 2, 1 by 3, 1 by, you see all the rational numbers over there, yeah? There is a small region which just, all the, it just attracts all the nearby frequencies. And so for example, if your initial omega, right, was let's say 0.11, right? So let's say it was 0.1. Initial omega is 0.1. And your k, the coupling strength is 1, right? Then after a little while, you will just attract both, both of them to have this particular, the frequency over here which is 0 by 1. Little thought will tell you it's the same as 1 by 1. So if the coupling is strong enough, even if the ratio of the two is not very close to each other, eventually they'll come close to one another. That is the entrainment. If your, if let's say your frequency, initial frequency is a half. So they are 1 by 2. And obviously your winding number is 1 by 2. But if it is half plus a small fraction or all the way actually up till almost 0.6 over here, let's say about 0.55, all right? If your coupling is strong enough, it goes exactly to 1 by, 1 by 2, okay? Yeah. Winding number equals to 0 is the same as winding number equals to 1. For the following reason, that on, in this modulo 1 algebra that I'm talking about, the 0.1 and the 0.0 are the same. Okay? So 0, so 0 by 1 is the same as 1 by 1. So that, it means identical. Yeah. You just keep on coming back to that point. Yeah. Thanks. Okay, now this is kind of important because also it's going to figure in a homework. I mean, dream up your function and calculate this picture, all right? That's going to be the homework, but I'll write it out properly. And if I find that too many people are dreaming of the same function, that will not be a good thing. Okay, but jokes apart, it's not good. Are you clear about what is happening over here? As you keep increasing the coupling, nearby frequencies get attracted to one another if they are sufficiently close to rational numbers. Now, you all know that on the line k equals 0, the number of rationals is of measure 0. Yeah? Almost every number to start with on the real line, if you pick a number at random, it is an irrational number. Yeah? Okay, most people and most numbers are quite irrational. Please, okay, so you're listening. All right. Now, by the time you get to k equals 1, all right, and this is an important line over here, but I'm going to give you the reference so you can read it. This is a beautiful three-page paper in physical review letters. You know, it's worth at least looking at once. Okay, so on this line, what's happened is that this ratio 1 is to 1 is now actually taken up a large fraction. The precise length of that can be calculated. It is 1 over 2 pi. The width of that particular zone over here is 1 by 2 pi. This is somewhat smaller, that's smaller and so on and so forth. You can see every rational number now has a little width around it, where it has attracted all the close by frequencies. On this line, the irrational numbers have measure 0. So, at k equals 1, no matter which omega you start with, you're very likely to end up with an entrained frequency. And for lower coupling over here, you can calculate something like that. It's easy to compute and the definition is trivial. So, you can calculate something and you can see what is the width of these. Yeah, so yeah. Okay, color white is irrational tori. And color light blue is, sorry, dark purple, whatever that number, that, aubergine. Okay, so the aubergine shading is for rational orbits, periodic orbits. Yeah? 1 is to 1 or is an orbit that just goes around like that. 1 is to 2 would be an orbit that goes around twice in one direction and once in one direction, like that. Yeah? If you took an omega of 0.9 or 0.8, no, not 0.8. So, if you took it at 0.9 and k equals 0.9, you'd find that the actual winding number is 1. So, this is an important diagram actually because this is a slightly more quantitative version of what Arnold did. And I just want to tell you that what happens above k equals 1 is that this map for the case of the sine circle map, there are non-differentiable points at certain parts of it. Red gives rise to overlapping, so these are called Arnold tongues because it was first described by Arnold when, and this is just for normalization of everybody, is in student work done in 1959. He was a student of Kolmogorov. So, this is what he's writing in a, it's a little difficult paper but anyway. He looked at this particular map. You can see that it's really the same map. I mean, just change some variables a little. Okay? And here he proves that the width of the tongue decreases as the power of the denominator of the, you see the point A. Sorry. Okay? So, A is the same as R omega, right? You know, without the 2 pi taken out. So, A is 2, M by N, where N is the denominator and the width of this, of each tongue, it goes down. It decreases as epsilon to the power N, epsilon being the coupling. And that's why these first ones over here, 0 by 1 and 1 by 1, they go down as exact straight lines. These are quadratic and cubic and so on and so forth. And these increasingly high powers means that the zones are narrow, narrow, narrow. So, actually this line at the top when K equals to 1 is just covered by rational windows. And there are gaps between them and those gaps contain irrationals. So, even at K equals 1, there are irrational orbits. Above that, over here, these start overlapping. Once they overlap, the dynamics gets really complex. Now, this is to, you know, wanted to just give you a flavor of why such, you know, this entrainment happens in nature. We don't, we almost always go for 1 is to 1 resonances or they are called resonances. But, you know, whenever you've got systems that are almost identical, then you think that the natural frequencies are the same or almost the same and then they get entrain. What this tells you is that if they are close to some small rational numbers but not identical, 1 by 2 or 1 by 3 or 2 by 3 or something like that, then also you can have a very good entrainment. And that can be done experimentally and we are living in at least one experimental system where this phenomenon of entrainment is very important. Does it mean that there are jumps in one of the directions in the trajectory? Well, it goes chaotic. When the tongues begin to overlap, okay, at k equals 1, this is called the devil's staircase. It's called the devil's staircase because at every rational number, there is a step. And if you step on each one of those rationals, it's going to take you and you're never going to get to the top. So you can see some of the big steps, but believe me, everywhere you just have step, step, step, step, step, fractal. Okay, so the system that we are living in of entrainment to which we are eternally grateful to external forces well outside our control, all of you know that you only see one side of the moon, right? Varum. I mean, this is an international class, 9. Okay, no, but I mean, certainly this is the kind of question that would have been of interest to people, right? Then why do we only see one side of the moon? I mean, if it doesn't keep you awake at night, no. Okay, so the reason is that the moon's, the reason why we only see one side of the moon is that with respect to the earth, I mean, we go around once a day. The moon goes around us once in something like 27 days, 27 points something, all right? But the moon also rotates on its own axis once in exactly the same time. Okay, so as it is going around, sorry, I can't do it, but okay, maybe we'll make this the earth and, right? So as it's going around, it just keeps rotating in exactly the same. As the moon takes 27.3 days, it just goes around in 27.3 days. So the lunar day, so to speak, all right, is the same as its period around the earth. And that's why we mostly see exactly the same part of the moon. Why we don't see mostly is because of Kepler, right? The moon's orbit, if it was perfectly circular, then we'd see exactly the same part, but it's not, right? And it's, then the tilt, there is a slight tilt of its axis with respect to our own axis. So sometimes we see a little more of the top, sometimes we see a little more of the bottom. But we don't see more than 60% of the moon anyhow. And the reason is that when the moon was formed, when the earth gave birth to the moon, so to speak, I mean, there was no fixing of angular momenta and so on and so forth, right? Entrainment for tidal forces to be somehow minimized on the moon as it evolved around over, I mean, we've been around for what, four billion years or so? I mean, not all of us, me closer to the moon, but we've been around for a long time. And the moon has had enough time to lose its energy to slow down until it just got into frequency locking, right? And this kind of tidal locking is there throughout our solar system. 20 out of 24 or 26 satellites that are known in the solar system are tidally locked, right? And Mercury is another story that has, it rotates 3s to 1 around the sun or some, you know, I mean, there are all these interesting little coincidences that come up, right? Okay, now I'm going to, whatever we have, you know, I have 15 minutes for this, but I really need to do a little more. So I will introduce the subject today as to what we'd like to understand. Is everyone cool and comfortable as far as circle maps, entrainment, et cetera? Ben? I'm sorry, it was very dated Italian. Okay, so now that we have an idea why these things do that, the sort of pulling of frequencies and so on, what we'd like to understand is how do we actually implement some of this in a system, you know, we started by discussing synchrony in the Huygens' clock, right? So what is the mechanism of this, right? I'm just going to say roughly what we are going to do tomorrow, because there's no point coming back to it later on. But this is a mechanical system, and at least the first approximation, it is actually quite simple. Let's say, right, so the mechanical system I want to describe is two pendulums of identical mass, identical length, suspended from a common beam. And this common beam is, you know, for convenience just got this particular property. None of these details is important, but as we discussed earlier, if I let the mass of the beam, which is capital M over here, go to infinity, all right? Then essentially the systems get uncoupled. So the Lagrangian for this system is, I mean, one just writes it down as one does with most mechanical objects. You've got the total mass of the system, and the base is allowed to move, all right? And you can think of having a spring along the base just to restoring one so that it doesn't slide off to infinity, all right? So here is the mass of the entire system and the velocity of the base. And here is the corresponding potential energy, if you like. So you've got that. And then you've got, here is the Lagrangian part which is just corresponding to the two pendula. You've got the, you know, kinetic energy for the pendulum, the angle theta 1, sorry, phi 1 and phi 2 is shown over there. I will probably call them theta 1 and theta 2 tomorrow, just whimsical. And this term couples the two. It's actually a talk, you know, from the, you know, it's a talk imposed by each pendulum on the, on this entire frame. So that much is your Lagrangian that you start with, all right? And, okay, so once you have this Lagrangian, you can write down the equations of motion for the two systems, that is the systems being the individual pendulum and the entire base. And, okay, so let me just run you through this. Sorry, if I look at the interaction term, it does not come from a potential. That's not coming from a potential. That is just the velocity of, it's a talk kind of a thing. I don't know what one would call it. But it's the velocity of your system that is moving and the velocity of the pendulum. Okay. I have another figure which I'll show tomorrow which makes it. So but you derive it from mechanical? Yeah, it's derived from first principles, so to speak. Yeah, all right. Now, ignore for a moment this term and this term. This equation of motion is got just from looking at this Lagrangian and obtaining the corresponding equations of motion for each one, Phi 1 and Phi 2, so k takes the values 1 and 2. For the entire system over here, again, ignore that particular term. I've just got that, all right. Now, what this, these two, these are actually three equations of motion. Two over here and one over here. This would be a conservative system because this Hamiltonian is conservative, all right. But we know that there is some amount of damping. There's phenomological damping that just comes from the fact that the pendulum is in air. So that is written as B times Phi k dot, all right. So this is just the, the damping on just some number that will be put in later on. Okay. I'm not getting into what kind of form it is because this is just a simple pendulum. If you actually try to model Huygens' pendulums because they were timekeeping devices and had to stay active in a sea voyage, the actual form of the damping is different and also there's a need for external forcing so that these pendulums just keep going. Otherwise, with just simple damping, it would just stop moving after a while, all right. So there are extra terms that are added. This is one extra term that is thrown in and this is the other extra term that is thrown in for the pendulums. And for the base, there is this extra phenomological damping friction, call it what you will, that is added in, all right. This will be the starting point for our discussion tomorrow. We will try to analyze this and see why the solution Phi 1 plus Phi 2 is equal to 0, all right. So you see now I've got two equations here, one for Phi 1 and one for Phi 2 and an equation for X, right. So I'm not going to be interested in the whole solution. That's something that I'll give you the references and you can look at. I just want to discuss the following situation. In-phase synchronization is given by the difference between the two variables. If I call a variable delta or theta or something, I don't know. Let's call variable 1 is the difference between Phi 1 and Phi 2 and the other variable has the sum of the two variables. Then we'll see that one of them goes to 0 and the one that goes to 0, if the difference between them goes to 0, then clearly the motion is Phi 1 equals Phi 2. Phi 1 minus Phi 2 is equal to 0. Contrary wise, if Phi 1 plus Phi 2 goes to 0, that says that they are exactly out of phase with one another. So aim of the game tomorrow will be just to see how we show that one solution dominates over the other. And this is complementary to the discussion of the sine circle map because the sine circle map is just all sorts of things just thrown into one. And this will be specific to two pendulums and possibly two metronomes. And with that, let's meet tomorrow. Sorry, Ram. So you should have momentum conservation of the whole system here, no? Which should be? Just one second. So can you see it clearly from this? See, what you would really like is M Phi 1 plus M Phi 2 dot plus M X dot is equal to is a conserved quantity. That's not easy to see because you've got both the input and the damping. No, but say I mean you just look at the Lagrangian. If you look at the Lagrangian, then...