 since I haven't done Italian before. We had started in the last time to deal with the general problem of chaotic synchronization. And the reasons are many, the simplest kinds of systems that one can solve all tend to be linear systems or simplified models of nonlinear behavior like the Kuramoto or whatever. And in general, most systems that we will encounter in real life, if not extremely chaotic, at least there are the kinds of systems where several kinds of behavior can take place. So it's important to understand synchrony in a somewhat general setting. This is just a picture I already showed you yesterday of two chaotic systems when coupled by this so-called master-slave geometry. After a while, the master takes over the slave completely. So the dynamics becomes identical. Now, this is a subject that has been studied for a long time. So there are all sorts of variations that are possible. And all of this comes under a framework which, for want of a better name, is called generalized synchrony. And the general scenario is the following. You have one equation, let's say for some function y, which is uncoupled to the rest. And the equation for one subsystem depends on y. So again, this is to be thought of as a master and a slave, if you like. But this is only a general idea, yeah? OK. The equation for the master doesn't depend on the slave. See, this equation for y is just a function of y and maybe other things. Did you get your umbrella back? Are you feeling it? You got your umbrella? It's in the main desk. OK, so this is a master only because it doesn't depend on any other variables. And that's a slave because that slave is getting input from the master. See, I mean, of course this is just mathematics, but I'm thinking of, let us say, cells that are coupled by some chemicals or what have you. I mean, I'm just imagining the kinds of situations you might like to apply this kind of model to. So cells mediated by small molecules, the production of the small molecules is independent of the cells, and that happens. Or you have a set of neurons, and they are driven by some calcium wave that comes from somewhere else. And that dynamics is independent of what happens to the neurons, but of course the neurons depend on what happens. So it's just that you call it unidirectional coupling, master's, slave, whatever. Now, so the idea is that for a suitable amount of coupling, there can be the so-called generalized synchronization when the response is a unique function of the drive. And this is just expanding your notion of what does it mean for two systems to synchronize. If they are identical, we can all recognize it. If they are out of phase, we can all recognize it. If they are related by some kind of mathematical transform, it's not always easy to recognize. For example, you can imagine this situation where here is your master, and the master has got some, maybe it's a chaotic master, so you've got some chaotic drive over here. You couple it to the system, and the response also looks chaotic. And this is quite to be expected, because we saw the example of a forced oscillator. When you force one oscillator with another, then you typically have overtones, and you have a combination of both the motions. So regardless of what the dynamics of y is, once you impose a chaotic drive on it, the response is going to be some funny thing. Now it's very difficult to look at this and say, because it's not identical to the master, you'd say that, aha, this is clearly synchronization. So what people have done over the years is to do the so- called auxiliary system approach. Namely, you take one copy of the master and one copy of the slave, and then you make another copy of the slave. If the two slaves are exactly in the same response to the master, then the two slaves will be perfectly synchronized. Not a particularly deep idea, but so you've got some complicated, so here is the example. Here you've got x as the master and y as the slave. And so when they are uncoupled, the variable on the x-axis is the master, the variable on the y-axis is the slave. So when they are uncoupled, you just get some, not entirely uncorrelated mess, because you can see something over there. But let's just say for small epsilon, you don't get any particular dependence over there. For intermediate epsilon, you get another function which looks kind of like that. But if you looked at the actual response of y, it would be quite chaotic looking, see? So this is the master, independent of y or y prime. So what I've done over here is to plot the master variable versus the slave variable. Both of them go in 0, 1. And let's say the coupling is very weak over here, but still there is no synchronization of any kind. When the coupling is very strong, the variables become identical, so it is completely synchronized. Here what I have done is to take slave 1 and slave 2. So slave 1 versus slave 2 is uncorrelated to start with. That's what that mess over there is. But here, when the master and the slave have got some complicated relation, the two slaves have the same complicated relation. So slave 1 versus slave 2 is on that diagonal manifold. No, it's slave 1 doesn't depend on slave 2, but they are both linked to the master. The response to the master is unique. I mean, the jury is not out on this. I'm going to give you all one problem, which is an open problem not to be solved necessarily this week. But those of you who would enjoy playing around with these kinds of systems, the scenario for generalized synchronization is not always master-slave. It can be bidirectional. I mean, to my belief, this is one of the most common kinds of synchrony that happens, because most coupled systems are not identical. So I'll just give you a problem. You solve it if you like. It's the best kind of problem, right? The main result that I want you to look at is this. You've got the master, slave 1 or slave 2 will both look like this. And if you plot slave 1 versus slave 2, they are identical, even though the dynamics looks crazy. You see, the assertion about generalized synchrony is that the response is a unique function of the drive. How do we visualize that? That it is a unique function. You make two of them and then say, right? Not the deepest idea, but it works. I mean, that is almost the point of the open problem that I'm just setting for you, OK? So this assumes a lot of interest in the kinds of systems that people who study complexity like to look at. Supposing I had two dynamical systems, but they were subject to the same noise, identical noise. Namely, I mean, so here we are all dynamical systems of some kind, right? And if we are subject to some immense amount of external noise, right? Would our response to the noise be the same, yeah? Now, this is actually the subject of a, so what Banawar and Maritan discovered in, I don't know, many years ago, and they had it published in PRL, was to take this chaotic system, which you'll have all seen, x is equal to 4x into 1 minus x. You know, this is the logistic map with maximum, right? And they added, they took two copies of it. Here y and y prime are written as x and y, so, right? They took two copies of it and added the same noise. So they had a signal which they generated, and at each step they added exactly the same noise to both the systems. And, OK, what does synchrony over here mean? The distance between x and y goes to 0. They become identical in that sort of thing. And what they showed was, here is a picture which I just took straight out of their paper, right? If the noise was identical, then the trajectories coincided after n steps, where n depended on the precision of the calculation, yeah? Gaussian. Yeah, stochastic noise. Because that depends on the precision of the calculation. See, the point about the precision is, this is a controversial paper with a lot of, OK, I mean, I'm just putting this out to you because when we say that you're being driven by a common system, think of this as two copies of the same system, and this eta is a drive which does not depend on x or y. It's the same scenario. It is a scenario of generalized synchronization. But I'm bringing it up because today when you do applications, or when you think about looking at complexity in such systems, having an external noise is a very important thing, all right? So, and here you see actually how the difference between the two goes rapidly down to zero, and the point about the precision of the calculation is that once you are down to zero, you will, if the distance between them goes to zero, it will stay zero forever. They can, all scenarios are possible. If you have the right coupling, then things with different functional forms can also show this kind of synchrony. By synchrony over here, we don't mean that the dynamics is identical. All that we mean is that somehow the correlations between the two systems are so strong that as I will, in the next few slides, I'm going to argue that this brings things onto a low dimensional manifold. Same area, yeah? Because you see, there are directions in which things diverge. In the Lorentz system, for example, there are three positive Lyapunov exponents. One is positive, one is zero, one is negative, all right? So, things will diverge along the direction of the positive Lyapunov exponent, and they will converge in the direction of the negative Lyapunov exponent, all right? So, if you don't have more than one positive Lyapunov exponent, all right? Generally, what will happen is that things will converge along this side, and they will diverge along this side. What happens in, okay, you see, when you've got these attractors which have got different shapes and the different parts of the system, you know that all parts of this attractor are not uniformly, they are not uniformly expanding or contracting. If you take, everyone has studied the logistic map, yeah, all right? So, if you take the logistic map, you know, so you've got xn plus one is equal to, let me just say this is f of xn, all right? Now, if I've got a fixed point of this map, let me call that x star, all right? So, let me take x star in a small displacement from that, right? n plus one, this is equal to f of x star plus delta, right? And this is just equal to f of x star, which is 0, right? X star is such that this is 0 plus delta n times f prime of x star. To first order, just first order Taylor expansion. And on this side, I've just got x star n plus one, which is the same as f of x star, so I cancel that out, and I've got delta n plus one. So, delta n plus one is just f prime of x star times delta n, right? So, depending on where x star is, right? This is telling us that the instability is just proportional to the slope. Now, the slope over here, right? Over here, the slope is, I mean, whatever, it's some positive number, right? When I come here, it becomes negative. So, the slope is some positive number. It goes to some negative, it crosses 0. At this point, it is 0. And then it goes down to some negative number over there, right? So in the entire region where the slope is from minus one, so this is slope one and this is slope minus one, this entire region is contracting. See, I'm just looking at deviations from x star. How do they propagate? They propagate as a function of the slope at x star, just Taylor expansion one. See, this is one term plus delta n squared f double prime by two factors, etc., etc. I'm ignoring all this, yeah? This one, because x star is a fixed point. So, x n star is just f of x n, those two just cancel out, yeah? On the proof about this, yeah? So what I want to say is that on any attractor, Lorenz, Rossler, logistic map, whatever, the slope is not uniformly, it's not unstable everywhere. I think when we were talking about Lyapunov exponents, people were asking, there are parts of phase space which will be pulling you in. There are parts of phase space that will be pulling you out, right? So it's not as if the phase space is uniformly attracting or uniformly contracting, all right? If you remember in my discussion of the total Lyapunov exponent, I said that this was equal to something like this limit of n going to infinity, one by n summation of i going from one to n log of f prime at i, right? This is the expression for the Lyapunov exponent because we were going, making many, many, many steps, looking at the expansion of the contraction, taking the logarithm and summing it and this was the expression. Now, it turns out that the Lyapunov exponent is the average rate of contraction or it's the average rate. So if I find, if I just take the Lyapunov exponent for one step, all right? And ask what is the distribution? It could look something like this, where this is the average Lyapunov exponent. But this is what the distribution of these terms looks like. So you can get a positive Lyapunov exponent, all right? Let's say this is 0, but there will be some parts of this trajectory, where the Lyapunov exponent is actually negative. And there are some parts where it is positive. And the average together gives you more than, makes it positive. But it's also likely that if you take your averages properly or depending on the trajectory in question, there could be different parts. I mean, the 0 might be here, in which case the Lyapunov exponent is negative. But there could still be parts of it which are positive. Meaning that when you're taking the sum of an infinite number of terms, it's actually the distribution that you have to look at. The average is what the Lyapunov exponent tells you, right? But if you look at the finite time variance of any of these quantities, those variances can be quite large. I mean, like today, we also look at situations where the change in entropy is positive, but there are parts of that entire trajectory where the entropy is negative. The change is delta s is negative, right? Jarzynski's theorem and all that, right? So the point about why these come together is that occasionally, this very occasionally, these trajectories go and hit a huge negative part. When they hit a huge negative part, they come together. And if they hit this negative part often enough, they come together and then they stay there. And how long they will stay there really depends on the precision of the calculation. If you're keeping 27 places of decimal and the difference is in the second or the third place, not gonna happen, yeah? All right, now the point is that you can therefore have noise induced synchronization. That is, you just subject some systems to the same noise and they will show this kind of synchronization. And okay, so this is just what I said. What people showed was that if you come to large contracting parts of the phase space, where does this apply? And I was just thinking of one situation which people do study. If you've got populations in different islands and the islands are close by and they don't allow migration between the two. Then because the climate itself is fluctuating and both populations of whatever species you care to look at, I think they were looking at sheep or squirrels or something like that. So you've got two populations which are unconnected to one another and they are subject to the same weather fluctuations. The populations of the two islands show very strong correlations. And you don't do things like put a predator on one and steal off all the sheep or something like that. Meaning you just have tried to do as similar things as possible. There will be variations in the climate even though they are very close by. But okay, so this is, I mean, this was an observation which has then also been now called an effect. It's called the Moran effect. So people do study the co-fluxuations of populations subject to the same weather or other conditions, right? So Moran's theorem states that a pair of spatially separated populations obeying identical dynamics in the absence of dispersals. So that is, you're not allowed to migrate between the two islands. And subject to random climate fluctuations will demonstrate a cross correlation that is equivalent to the correlation of the climates, okay? In some senses, okay? I'm using, no. No, it's, see, they observed it in Scotland and wrote a paper in nature and that was it, yeah, yeah, yeah, yeah, yeah, yeah. Because I mean, that's what the obeying identical dynamics, right? We have sheep on one end, foxes on the other. They're not obeying identical dynamics, right? So they also studied these blue tits and other such things, right? Now, I know that today when we study complexity, biological systems are among the more interesting kinds of systems where synchrony, etc., is to be observed. And I just want to point out that biological rhythms range from fractions of seconds, microseconds, less even. All the way up to years. So there are ranges of time scales and then worrying about how these are all coupled to one another is a problem, right? So neural rhythms up to ten seconds and below that even, right? Cardiac rhythms, typically for healthy hearts is about once a second. Calcium oscillations, the main thing in communication between cells, seconds to minutes, biochemical oscillations, etc., etc. So and the rhythms in ecology and epidemiology go over years, right? So the idea of quote unquote synchronization has to be expanded to also include these various time scales. Now, there is a whole branch of this called stochastic synchronization. Namely, if you've got systems that are already intrinsically noisy, not subject to external noise but something which is intrinsically noisy. Let's say cells, reactions within cells. Then how does, how do these synchronize? I will come to this a little later because I want to get this idea across. What we've already seen in the pictures that I've shown you about chaotic system synchronizing is that once you come to the state of synchrony, you are on a lower dimensional sub manifold. Just take it as a lower dimensional subspace on which all your dynamics is occurring because if you've got two Lorentz systems, one Lorentz is three dimensions, the other Lorentz is three dimensions. And the total dynamics takes place in six dimensions. But finally you come down to the case where x1 is equal to y1, x2 is equal to y2, x3 is equal to y3. And that defines a three dimensional subspace. So the synchronization manifold is three dimensions inside a total of six. Now, if you want synchronization to be stable, perturbations away from this manifold should bring you back to the manifold. Because once you are synchronized, as you asked, how long will they stay together? In the case of the Banawar-Maritan business, it depends on the precision of the calculation. But basically what's that saying is that if your perturbations away from this manifold are very strong, then you can come on to this manifold and then after a while you'll drift apart which may or may not be useful to you. So I'm making this proposal that any coupling that brings the dynamics to lower dimensional subspace is some form of synchrony. Now this, in generalized synchronization, the formalism, one second let me just try to, okay? So what I'm going to, okay? So I have one dynamical system, this underbar means it's a vector, right? This is some f of x and it may depend on x alone. Y is another dynamical system and this I'll indicate as y, right? Now this basically means I've got two dynamical systems which don't even need to be identical. Now I want to couple them zeta x which may depend on both x and y and zeta y which may also depend on both x and y, right? And my objective is to get that y is some unique function of x, okay? These are two coupling and this is this unique function. Now what we've done in this earlier example, for example, this one over here, okay? What we've done in this particular example is to say this x is f whatever, this is f of x, this is f of y. And we said that this term was 0, zeta x is 0 because there's no term over there. And zeta y had both x and y in it. When we couple them like this, then we eventually saw that although the relationship y is equal to phi of x was very likely to be non-differentiable. And you know, it's complicated. I mean, look at the middle line over there, sorry, over here. Like this is not a simple relation between x and y. However, it was unique. Now in the approach that I want to describe, in the approach, okay? So when this function is smooth and when this function is non-differentiable, the terminology in the field is that this is called weak generalized synchronization. And when it is differentiable, it's called strong generalized synchronization, right? And what I want to propose is that, supposing I have a nice, supposing it's a differentiable function. And I specify this, given this, can I determine these two? Namely, can I determine the coupling that is guaranteed to give me this? For example, if I want y to be equal to x, sorry, if I want y to be equal to x that is the case of perfect synchronization. What is the coupling that I have to add to the two of them? Yes, I know I have introduced several types of coupling that give you identical stuff. But is there a general framework which says, if I want a particular kind of relation between these two, can I deduce the coupling that I must add, right? Namely, can I reverse engineer this process, okay? So for that, let me say that this functional relationship, the notation is a little messy, but please bear with me, all right? The idea is to specify the relationship that I want and deduce the coupling. And if I take the two variables of the system like so, and this is your functional relationship that is required, then this comes out to be a set of several conditions. See, for example, if I want perfect synchronization, then phi 1 of x, y is x1 equals y1, phi 2 is x2 equals y2, phi 3 is x3 equals y3 and so on. And that will give me this condition of perfect synchronization, okay? I'm going to be giving you these notes so the algebra is something that I hope you will work out. And the idea is the following. Supposing I have a surface inside of phase space, and I want my dynamics to be on that surface. This surface is the synchronization manifold, and I want my dynamics to be on that surface. The main idea over here is that I would like my coupling to bring me to that surface, and then stay on that surface. So the way in which this is done, you bring nearby trajectories on to the surface. Let's say the surface is the plane, and then make sure that the motion stays on the plane. If the motion is staying on the plane, it has to be orthogonal to the normals on the surface, right? So at every point in the surface, there are normals. If I want my motion to stay orthogonal to that, that is done by the coupling. And then it frees me away from what kind of surface I need this to be. So long as I can define normals to the surface everywhere, okay? So the algebra is not particularly difficult, okay? So in order that this submanifold specified is invariant, a trajectory that is in this submanifold will remain on it. So we do that by asking that the flow direction be orthogonal to the normal vectors at any point, all right? So now the normals are given by very straightforward. The normals are just given by the, you know, the derivative, the divergence at that point over there, right? And if you define all the normals to be this gothic n and the flow equations to be that, this is just the flow equations. I've got these normals. And the requirement is that the flow direction is, you know that the cross product of the normals and the flow is 0. Then it, then, and you add the coupling zeta. You find that this is the equation that has to be solved. Now this is not unique because the dimension of n, n is a, in this particular case it is a 3 cross 6 vector, right? Zetas are six dimensional. So these, this set of equations gives you many, many sets of solutions. Underscoring the fact that synchronization is not a unique problem. You've already seen examples of it. I showed you that Lorenz, Lorenz was getting synchronized with master slave. Then it was getting synchronized with x1 minus x2. Many forms will give you the same synchronization. But now that you have an equation like this, this is a matrix equation. And this is undetermined. So there will be many, many forms of coupling that, this is the coupling matrix over here, right? So many, many forms of coupling that will give you the same requirement. But all of them will not be equally stable. So because they may not all be equally stable, you can add stabilizing terms. Namely you can add some extra term which vanishes on the manifold. So if you're adding a term which vanishes on the manifold, then it doesn't, again, it doesn't affect the dynamics, all right? Okay, so this is your, this is your target surface. And how do you bring it? Probably easy to at least demonstrate how this happens with one example, all right? At every point on the submanifold you want the eigenvalues of this Jacobian matrix to be negative. And you can arrange that by certain amount of art. But let me, let me give you an example of how this works. If I've got two copies, two Lorenzes, okay? These are two copies of the Lorenz which I've just written it in X1, X2, X3 rather than X, Y, Z, okay? The condition that I want on my submanifold is not X1 equals Y1, X2 equals Y2 and X3 equals Y3. But some scaled version of the, of the three. So X1 is equal to A times B times C times. So this is like taking this diagonal synchronization manifold and just flipping it around or doing whatever you want with it, okay? Now, this is my F of X, this is my F of Y, this is my Phi, that is the three constraints that I want. And I put it back into, so I put it back into this equation over here. I know what the normals are, I know what FX is, and I know the normals over here. The thing to do is to determine what is zeta, what is the coupling, yeah? You've got the equations of the plane, right? See, this is your manifold. So it's X1 minus AY1, X2 minus BY2, X3 minus BY3. And at any point X, Y, Z, you can figure out the normals, all right? Okay, so, and that's the answer. One particular solution that was found was that zeta1 is that, zeta2 is that, I mean these are messy, there's nothing intuitive about it. But if I take A equals 1, B equals 2, C equals 3, that is, I just take the Lorentz and I want to blow it up. With that coupling, that's the answer. So you can see that, and when you sort of plot it in the right dimensions, you will see that X, I mean that this manifold is not the usual perfect synchronization manifold, this is what's called projective synchronization. So now it turns out that there's a lot of choice that is available. I could choose, in fact, zeta1 over here to be 0, 0, 0. In which case, X would be the master and whatever comes out on this side would be the slave. Namely, this formalism that I have over here, namely, sorry, you know, namely to find the coupling given all the others that are known. Because it is non-unique, I can just choose a variety of different kinds of coupling forms, right? So here is a, it also, I should tell you that it doesn't matter that the systems have to be identical, they can be different and so on, exactly. No, this is not the, I'm going to show you in this next example, I'm going to show you three couplings that do the same job, right? Okay, so here is another objective, let's say X1 equals Y1, X2 equals Y2, but now X3 squared is equal to Y3, right? So this is a surface which is no longer just flat. This is a surface which is at least in one direction, just curved between X3 and Y3, yeah? So again, you go back and put down all those equations, calculate the normals, and that's the answer, all right? So I made this X system the master, I made the Y system the slave. And so the master's coupling is zero, the master's not coupled to the slave. And here's the slave being coupled to the master in whichever some complicated way. And you can see, I mean, visually, two things over here. One, of course, that the third variable is just a square taken up there. And I have plotted this on a subspace. And you can see the curvature that it is following this X, what is it? X3 squared is equal to Y3, it's just a simple parabola over there. And this is the coupling that does it, all right? And to ask, is this the only coupling that will do it? No. I can, instead of taking the constraints this way, I can say that X3 is equal to the square root of Y3. It's the same constraint, right? And then I can make Y the master and X the slave, if you like. Or I can say let's go for bidirectional coupling, right? So if I make Y the master and X the slave, that's the coupling that will bring it on to this blue one, I think, right? And if I say that let the coupling be bidirectional on both sides, then I get that coupling and I get the red trajectory, yeah? This is very particular to Lorentz because Lorentz is always positive in the Z, okay? Because both Zeta1 and Zeta2 are non-zero. See, this is unidirectional because all the coupling terms here are zero, right? I mean, this is lots of algebra. But what I want to point out is that once you've got the manifold, you come onto the manifold and you're stuck there. But you could be stuck in different parts of it. They're all in the same manifold, right? And they all have the same relationship that X3 squared is equal to Y3. But they're on different parts of it, right? And this kind of reverse engineering is not only it's possible, right? Because, I mean, you can ask whether this is a natural coupling or not. But you know, we've got to go beyond just X1 minus X2. We need to sort of expand the idea, especially in neuronal systems. Your couplings are crazy. I mean, I know that people are interested in such systems. And the coupling is not always a simple diffusive coupling or nearest neighbor or what have you. So I'm just alerting you to the fact that couplings can be a zoology in themselves, all right? So this works, this works. The earlier one also, this also works. All of them will bring the dynamics to the same manifold, submanifold. And all of them will keep it there because they are actually quite stable. Yeah? Yes, sir. See, it's coming. No, it's a matrix equation. N is a six by three matrix. I mean, this is just go to, I'm going to give you the references. There's a paper we wrote in this review, which sort of, it is the normal of the service, it's just going to take it in all directions. So then it comes, you know, sort of the big matrix which comes out. Big one? I'm not sure, but we can discuss this. Meaning it's just sort of very straightforward mathematics and linear stuff. It's just that you come up with a set of equations which have got more unknowns, sorry, more constraints than unknowns or some such thing like that. So you finally find many, many solutions. And I'm going to show you an example, both, yeah. No, I haven't, I haven't A thought about it in that way that you go from one solution to another. But no, I don't think there is a protocol. See, there are, depending on your applications, there are some solutions or some couplings which are natural and some which are unnatural. To give you an example, supposing I wanted to implement this on a electronic circuit. I make one circuit with Lorenz one and make another circuit and how do I couple them? This is actually a very good coupling to do that. For this, I mean, this is just hell because I've got X1, X2, X3 divided by Y3. So these are in a sense not very natural couplings. They're just mathematical. But the fact that you can have many different couplings should in a sense inspire, okay? I mean, okay, all right, I mean, the thing is also that when people present work on synchronization, it's almost always like this. I've got this one system, I've got another system. I couple them like this and see they have synchronized. But the question is really, is this the only way to synchronize them? The answer is no. Infinite number of couplings will give you synchronization. I'm now thinking from the next step onwards that can you engineer synchronization that has just given some arbitrary systems? Can you make them synchronized? This is one way of at least. Simplest ways, of course, to couple them with an infinite strength. And then they always synchronize. But what you want to do is something a little more subtle and less invasive. So this is one way, okay? All right, so when I started my lectures, I thanked Matteo especially for a particular question. I had presented some of this at a meeting. Matteo was in the audience. And he said that, can you do something to look at spatial freedoms? For example, something like that, okay? So that is a picture of birds, okay? So this is a flock of birds, about a million or so. And there is a photographer in Barcelona called Shavi, who has now become a very good friend of mine. Because I keep on stealing his pictures. All right, so when you take a set of birds like that, what Shavi does is to focus on the flight of a bird, videograph it, and then superimpose stills from that. So you can see over here, the wave pattern, i.e., an oscillator moving in three-dimensional space. And here's a whole bunch of them, right? And when you look at them a little closer, see there are different heights. That's why the trajectories seem to be crossing. But as you see them, I mean, you just take this pair. I mean, the two oscillators that are there are definitely in some kind of synchronization, right? And I mean, you look at various pairs of them, you've got these. So what you really have are moving oscillators, right? And they have to have a certain relationship of synchrony, otherwise they're going to be colliding into one another. Anybody who's watched these huge flocks of birds, they die because the predator will come, and so that's the predator over there. You can see that predator coming in, hawk swooping through that bloody flock. And I mean, all these have escaped, all those have escaped, but there's one sad chap over there who's gone, right? But they don't kill themselves by colliding into one another, right? Unlike Tesla cars, okay? So the question is that what can we do with this, all right? So I will come back to discussing flocking in another lecture, but I just want to show you one particular thing, all right? Now, what you want is the same objectives, your uncoupled equations are that. You want to do the coupling like so. It turns out to be a very simple way of making this happen. The condition is you want pi of x minus y is equal to zero. So if that is my condition over there, right? And I've got x dot is equal to f of x. Let me already put psi of x equals zero. So that's going to be the master, all right? Then I've got y dot is equal to that equation over there, given psi x is equal to zero. Let me choose, let me choose zeta of y is this quantity. Where j is this Jacobian, and that eventually gives me this following equation. I mean, I just put in all the various variables here and there. And eventually, all I get out of this is that d by dt of phi x minus y, this is the constraint that I want, is just equal to minus r times the same constraint. Which means that phi x minus y is exponentially going to go to zero. Now, what happens is therefore you've got x dot is equal to f of x and y dot. Y dot has forgotten all about f of y. You see, we started with x dot is that and y was f of y plus something. It's completely forgotten. Namely, you can make these things happen if you completely destroy the dynamics of the slave, all right? But there is one silver lining. If the constraint is linear, namely if phi of x is x plus c, then you get the following sets of equations saying that the master moves with one velocity. This is after all the velocity of x, right? The slave moves with the same velocity. And in the flock, they're all flying at essentially the same speed. Otherwise, they're going to be colliding into one another. And they keep, and this is your constraint. This is your additional constraint, okay? I mean, this is your coupling constraint. Call it what you will, all right? So if you do that, then the following emerges. First of all, I just do as an example of two van der Poel oscillators. Here is one van der Poel oscillator, here is the other van der Poel oscillator. And I just want them to be separated by a distance a in x and b in y, all right? So x2 minus x1 is equal to a, y2 minus y1 is equal to b. That is, I want them to have the same dynamics. I just want them to be apart from one another. And that is, I hope you can see this green thingy over here. So the two of them are just separated from one another. They're both going around the phase plane. But, and this is the coupling that I can, if I go through the entire algebra that I've been talking about, that is the kind of coupling that I have. So this is the master, that is the slave, and so on, yeah? So it's actually possible to put in a constraint which is just purely spatial. Now, birds are not flying in van der Poel loops. So supposing I have the blue trajectory as just some trajectory of the birds that are flying by, I can make any other bird follow exactly the same trajectory. In fact, I can make 20 of them follow the same trajectory. Once again, I mean, this is, again, just mathematics as one master, that's following some trajectory, all the slaves, same velocity, keeping the distance fixed from one another. There's even a little noise being added over here. And you can see how they all come together, yeah? It is just now that the spatial synchronization, I mean, it's the same formalism. It is that my constraint, my constraint is just that the distance between the two is a constant number. I know we did, but well, I mean, and I'm saying that if some of the degrees of freedom are spatial, then it turns out that corresponding to those, all I really require is this sort of this linear, that has just this linear part to it, right? I mean, this is not as if it's a new thing. It is just an application of the old formalism to a situation where that's the only thing. And then you have even simpler conditions, right? And because it's simpler and you can make this happen, we said, all right, let's take some drones and put it on there. It's not the greatest demonstration, but you can see. There's an intruder who comes in. That's an intruder, not being controlled. But okay, the point I really want to make is that you can put, you can implement this algorithm in sort of autonomous vehicles and then make one of them the master. All the others are following specified distances from the master. And this is in three dimensions, so everything sort of, there's a little aberration in your visualization because we had to take the pictures from the ground and this is up in the air. But basically you can see that they all move together in the required constraint, yeah? One of them, only one of them is controlled. And every other drone is taking its instructions from the master. See, over here, we had the dynamics of the slave having some dynamic. Uncoupled, of course, it was just the dynamics of the slave and then there was a coupling with. It just turns out that that entire procedure just works. I mean, it's just algebra over here, right? But also if you think about it, if you've got two objects that are moving. If this one is moving very fast and this one in order to keep a spatial distance, which is fixed, has to move with the same velocity. If it doesn't move with the same velocity, then the distance between them is just going to keep increasing. So if you want the distance to be fixed, then you've got to make sure that the velocities are the same, yeah? Absolutely, see the point is with autonomous vehicles, you've got to control it somehow, otherwise you're going to be having collisions and things. I mean, my initial idea in doing this work was to actually try to understand whether we can put in these kinds of just simple linear constraints and make things work, right? And then you think in terms of drones, you want them to, well, the applications of drones are not particularly nice these days. But you can imagine wanting many drones to go and deliver some target, right? Sorry, well, right now we have not put in any interaction between the followers. But we have just said that if the leader has specified relations with the followers which are non-intersecting, then it shouldn't matter. But one can imagine doing more sophisticated stuff. In particular, it was my tragedy that I worked with a company that was more interested in drone shows than anything else. No, if you're looking at drone shows, you want to make money, and which is not the most evil thing in the world. But the trouble with drone show people is that they all work with 200 gram drones. And with 200 gram drones, there's a little gust of wind and the drone is flying off on its own, so I haven't given up completely. But no, okay, no, the point is that this may, or, so that was the end of. See, the point is that these may or may not be applicable to just one second. Let me try to get back to that one slide that I wanted. Okay, you can see that the spatial separation between different elements of this is essentially fixed. And they're all moving around some, I mean, I don't know whether there is a single master or not. But there is something which organizes all of them into this kind of very almost like a lattice, right? They're all moving on that. And I know it's a step between this and drones that are designed to keep a certain distance apart and so on and so forth. But one of the applications we are working on now is to say that what of the distance between any of these two objects is a pre-specified function of time, right? So if you've got that the distance has to go as cosine omega t, can you have these drones going in and out like that? Or this again is, of course, I mean, it's huge and beautiful. But there is another point over here. Uncoupled, a million birds in configuration space. What is the dimension of that configuration space? Three million, right? Each one of them is three. Now, this object, whatever it is, does not have three million degrees of freedom. They're all moving together with a constraint. And so one would find that the synchronization manifold over here is not three. But it's not something like three million. It's probably even allowing for all the variations in between. I don't know whether it is much, much more than about 20 or 30. See, if they were all moving together, it's a rigid body. It would be three. Okay, so this is a deformable body, but it's certainly not as of all the birds are flying around. So we are coming down to a lower dimensional submanifold. So in the language that I would like to say that yes, there is some kind of synchrony happening over here. It's not trivial synchrony of the X1 equals X2. It's not even the simple one that I showed you with the drones where distances are fixed, right? Because there again you've got each one of the drones has got three freedoms, constraints bring you down to nine minus some six constraints or something. So clearly there is, I mean all theories of this Parisi onwards don't have a master and slaves. All they say is that each bird looks at seven birds around it and then goes to an average distance, etc. My point more is really to say that when such a thing happens, the dimension reduction is something very important in this case, right? And dimension reduction, like course graining, is a way of going from, what I've done is to do the dimension reduction in the dynamical systems point of view. From a statistical system, all you say is that seven around, but there is no constraint that is put on the distance between the birds. They're all supposed to have the same velocity or some such thing like that, to start with. We check model, I don't know, some of you might have studied this. We check model or similar such models of flocking. I will come to it in a later lecture. I hope you'll notice that I will be giving my lecture on Wednesday afternoon and no lecture on Thursday. Because you have an exam in any case, and then Friday is our exam. I've been requested to stop my class a little early today because of the CSI entrance exam. Now that the class is a little fuller, who is taking the CSI entrance exam? One, the others are not here. The others are already taking that. Who's David? You're David? I've always thought of you as Lao. Okay, okay, so we'll stop today. And tomorrow we'll come back to other aspects. Is there even a cool, I'm not going to come back to generalize synchrony unless you want to have a private discussion sometime, we can do that. But you're cool with the ideas of chaotic synchronization, generalized synchrony and so on, yeah? It does, I mean, that's why it is a controversial topic. It depends on the variance of the noise, it depends on the intensity of the noise and so on. So, I mean, there are some people who don't believe in it at all. I introduced it only to tell you that so long as two systems are subject to the same kinds of fluctuations, there are situations when they become completely correlated. That fluctuation could be chaotic signals. It could be periodic signals. It could be quasi periodic signals, in which case you have something totally different and interesting, and it can even be noise, yeah?