 Voila. Bon dia. So today's language of greeting is Portuguese. All right. I thought I would continue from where we left off yesterday. See, the Lorentz equations were written down in 1963 or thereabouts. And those were the days of very slow computers and so on and so forth. But since in the last 60 or so years, there's been a huge, huge amount of both analytic as well as computational work on a variety of the Lorentz system and various other attractors, which I will talk a little about today. And the Lorentz is like the hydrogen atom of chaos explorations. So a lot is known about it, partly because, A, it's quite simple. And the other is that, you know, as you build up more experience, there is a lot more to understand about the system. All right. So the idea of attraction to some set, which is not a point, not a circle, on which the motion is chaotic, mainly that it shows sensitivity to initial conditions, this is like a crucial idea in complexity. That different initial conditions still give the same final result. Then there are lots of ramifications of that. But the system is simple enough, so one can actually try to look at it from what we were doing for two-dimensional systems. You can find the fixed points. There is one fixed point always, which is stable up till the value rho equals 1. And it will occasionally switch between rho and r. So the second two fixed points, these are two actually, they exist only above rho equals 1 on the real axis. So the behavior of the system has largely been explored as a function of rho, keeping other parameters fixed. And this is a schematic of a very complicated bifurcation diagram. So as you vary r, that is, if you're varying rho, initially 0 is the only fixed point. At r is equal to 1, there is a bifurcation making 0, 0 becomes unstable, and these two fixed points, they're called c plus and c minus, they become stable. So the idea is that there is one branch which is going out from here and another branch which is going out below. I'm just not, this is the symmetry in the system because there is this, the Lorentz system has the symmetry that it is invariant under this transformation. X, Y going to minus X minus Y, Z being, between 0 and 13.926, which is actually a precisely calculable number, there are only fixed points in the system. There's one unstable fixed point and two stable fixed points. If you look, the two stable fixed points have got different basins of attraction and 0 has, I mean, 0, of course, is fixed over here. So the whole of space just collapses into these two fixed points. No matter where you start, you'll go there. Which this one? Yeah. Or just algebraically evident. The y side, though, the rule of change. No, Z is not, you see, it's X, Y, Z going to minus X minus Y, Z. All right. So at 13.926, there is a bifurcation leading to the following situation that there is an unstable limit cycle and this point continues to be stable. So there is this bifurcation that happens over here and this unstable, sorry, the unstable limit cycles, they are actually born out of an unstable fixed point, which then becomes stable. You can see the dashed lines are all unstable. The solid lines all indicate stable. And this is called the hop bifurcation, namely when a stable fixed point crosses a certain, at a particular point, as you change the parameter, the stable fixed point becomes unstable and there is a stable limit cycle that is born. I will consider the Stuart Landau system a little later and I will show you how that happens on the board. So that is actually reversed over here. So you have this fixed point stable up till this value of whatever, R, H for hop and then that becomes unstable. Now, for this particular system, it turns out that up till this value over here, 24.06, I mean these are both important and not important, meaning other systems will obviously have very different kinds of bifurcation diagrams. They will have a whole different zoology to look at. But you have a strange attractor which is born at 24.06 and that goes on. But for a very brief period, the strange attractor coexist with two fixed points. Some initial conditions take you to the fixed point, some initial conditions take you to the strange attractor. And these have very interesting topologies to look at. As I've already told you, this is the values that Lorenz settled on and these were really nice because R is equal to 28. There are no fixed points that are stable. You only have the strange attractor. And if he had chosen a smaller value, there would have been a more complicated kind of... I mean it would have been very difficult to get so much interest in this business. His whole idea was to try to prove that the weather is aperiodic and therefore not predictable, where the forecasting is not an easy science. But I mean there are many things, if you plot them in three dimensions, they settled on to a complicated state. But this strange attractor is not a point, it's not a curve, or even a surface, it is a fractal of some kind. The word fractal is okay with everybody? Yeah? No? Okay. Monday. Small introduction, but Monday. I mean right now I want to do something else. Okay. Yeah. Why there are... What is an unstable limit cycle? Like an unstable fixed point. Okay. Let me look at the following equation. So if I look at this particular... And okay. So here obviously there are three fixed points for this equation in R, which is R is equal to 0, 1, 2. Yeah? R is a radius, always positive. Yeah? Yeah. Is it just by construction? So now if I've got R, if I'm just looking at R, this is 0, this is 1, and that is 2. These are the fixed points just along the radius. Right? Yeah. I'm sorry, I come up with these slightly silly examples, but you can expand it. Yeah. Okay. So here you've got a trajectory which is on some surface. Okay. So let's go just by elimination. Is it... Can it be a one-dimensional... Can the trajectory trace out in one dimension completely? No. Yeah? Important point is that at any point there can be one and only one trajectory. So even though this curve seems to be crossing and crossing and crossing itself all the time, it cannot be crossing itself... It cannot really be crossing itself. So the surface, you know, this flat thing that looks like the wings of a butterfly, right? There's actually many, many, many, many... I mean, there's an infinite number of layers over there. The trajectory goes back onto the same parts of space but without ever intersecting. No, we're looking in 3D. I mean that's perspective 3D. This is a perspective diagram in 3D. When you look at it, this trajectory is never repeating and it is occupying the same part of that space. Right? Yeah. So one thing we know is that it cannot be a two-dimensional surface because if it was a two-dimensional surface it would have to cross itself. You go... I mean it's many steps from there so it is a fractal but I'm just saying it is a fractal. It has that fractal geometry which is why it allows the trajectory to explore many, many... You know, if you cut this in a straight line, right? If you just cut one of those wings of the butterfly by a straight line, you would not see two points only. You'd find actually many, many layers. Yeah? Exactly. Exactly. And then you can measure and whatnot. Some of these things are very difficult to prove mathematically actually. So self-similar... No, you shouldn't. You're seeing it. You know, stop taking what you're taking. It's fractal in the way in which, you know, if you cut it, then when you start looking at it you'll find that it's all going on different layers. Okay? All right. See the last part of it, the Lyapunov exponents can be calculated for this system and as I told you, I just asserted yesterday the three of them. The three come because of the following fact that there are three equations of motion over here. So this entire system starts out in R3. Then there is an attractant. It brings you down to zero volume. Why does it bring you down to zero volume? It's because you can see that the divergence of this vector field over here, right? You've got that vector field over there. You can calculate its divergence. It is just sigma, sorry. It's minus sigma, minus 1, minus beta. Minus sigma, minus 1, minus beta. The sum of those three terms is the divergence which is that. Now in a theorem which has been proved by, you know, a long time ago, the sum of the three Lyapunov exponents has to be that. Right? So it turns out that one of them is positive so long as rho is bigger than 24.74 something. All right? That is, sorry I should, okay. So at least one of them is positive from this point onwards. That is a whole nother story. I mean it is something that for a very, very long time is unrepeating and so on and so forth. And then I'm not even sure what finally happens but that's how long it goes on. Okay. Another system, yeah? Because you're working in three dimensions. No, I didn't say that. There are three Lyapunov exponents because you're working in three dimensions. But it is not as if you can associate one with X, one with Y, one with Z. What happens, I mean what you can show is that you must have three Lyapunov exponents. That's one assertion. The second part is that because there is only beginning now, there's some understanding of what these Lyapunov dimensions are, what are these three of them? Because the transformation between that and the XYZ is not simple. So it's not as if in the X direction if a positive Lyapunov exponent, you're blowing up an X, that's not how it happens. Again, in the Lorenz system, all these three coordinates are all mixed in one into the other. So it's not as if you have evolution only in X, only in Y, only in Z. So the, what's the definition of? Okay. Okay, so a little time out before I come to the other system on the board. Practically how you study this system, you've got these three equations, which gives you the orbit, the ones that I've been writing down, sigma Y minus X, whatever that is. Now, the Lyapunov exponents are computed as I was trained to give you schematically yesterday. You start with the unit vector, right? And see how that grows along the trajectory, right? So to calculate the unit, the, and this unit vector is, you know, is, it is estimating the slope at each point. So it's actually going along the tangent, right? You've got the trajectory itself and the growth of this unit vector that you have is determined by the tangent with some tangent space. So for that, you look at this equation, delta X, delta Y, and delta Z. This is just looking at the linear motion. So if I, see, let me write down the Jacobian. It's minus sigma, sigma 0 and the ZX, what's it? Minus Z plus, I'm sorry, minus Z plus R. It's 1 minus 1 and with Z it is X, right? And the third one is YX and minus beta. So this is the Jacobian. I've just taken, you know, just taken, if I call this F sub X, it's partial of FX with respect to X, partial of FX with respect to Y, partial of FX with Z, and so on and so on. So that's, and then you fill in the others and you get a matrix like this, okay? So what I'm going to be interested in is looking at the dynamics of displacements where delta X dot, delta Y dot, and delta Z dot is given by this Jacobian times, I mean some of this must be familiar, right? Okay? So now I have to start with a unit vector. So this unit vector could be 1, 0, 0. It could be 0, 1, 0 or 0, 0, 1, any of these. So what I do is to start with all three of them. So this is my equation for the Lorentz. Now I come up with three evolution equations, three sets of the tangent vectors. So here I get, let's say the first one, then similarly for the second and similarly for the third. I mean there are many ways of doing this, but I'm just describing the way in which it's computationally the most straightforward to look at. So I start with three unit vectors and as they move along the trajectory, they will change from being the three orthogonal ones to some point over here. At that point you ask, right? What you ask is how much did they change in the direction X1, in the direction Y1, in the direction Z1, etc., etc. Then again you make them, you use Gram-Schmidt, the Gram-Schmidt procedure, you bring them back to length one. But now they are not pointing in, let's say in X, Y, they're not pointing in this direction. They're going to be in some other directions. Then they evolve around the trajectory again. And here again they have changed. Maybe they're looking like whatever. And then again you normalize, bring them back and you follow the trajectory as it goes around, each time going a certain length, seeing how much the length has changed and so on. So when you do that, you will find that there are these three quote unquote directions and initially of course it started out as X, Y and Z. But because of the way in which the evolution takes place, you find that they are just three of them, not particularly with X or Y or Z, because the flow mixes these directions up. See sometimes I mean staring at equations and trying to calculate it is a good way of doing it, of trying to get some intuition about it. So the main point is that if there are only three Lyapunov exponents and one of them is positive, then everything is going to expand along that direction, but that direction keeps changing as you go around the trajectory. Here is another direction which is severely contracting because the divergence of the whole vector field is something like this. It is e to the, what's it? The volume will go as volume naught times e to the 1 plus beta plus sigma times t. And for the parameters that you have chosen, it's sigma is 10, so that's 11 and beta is 8 by 3. So those are the ones that I said we use the calculation. So it's something like 13. So at each time step the volume is decreasing exponentially by a factor of e to the minus 13. That's why the volume is going to zero, but it could go on to zero if it was going into a two-dimensional plane or onto a one-dimensional line. But you can see that it cannot be a two-dimensional plane. So this is sort of the natural place where fractals start making their presence felt. Now there's a simpler set of equations called the Rosler equations which some of you may not have seen before. I mean, Lorenz I think everyone has seen, but Rosler is not quite so common. Although it is much simpler, there's only a single, there's a single term which is nonlinear and that is the xz in the third equation. You have xz. Unlike the other systems we've been looking at, 0, 0, 0 is not a fixed point. There are two fixed points for this system and one of them is the one marked in red over here. And there is another one somewhere out there that doesn't have any interesting dynamics around it. What is nice about the Rosler, and here's this part, the flow actually is not confined to a folded two-dimensional surface. Rather to a folded disk of finite width. There's been a lot of work on the Rosler systems, even though it's relatively less familiar. Now the point about the Rosler is that if I view it from above, that is if I project everything down to the xy plane, that's what it looks like. The motion is always circulating about this point. Of course, this is just a flat projection. It's always circulating about that point. But it's chaotic, even though it's circulating about the point. It is chaotic and I just thought I would show you one little demonstration. Two initial points that start out close enough but slightly differing in z. You can see that the way in which, hopefully this will also give you some intuition of Lyapunov stability. This is not Lyapunov stable. So we started out together, but then the trajectories are really evolving differently. But is it okay? No. See the point is that if you choose your epsilon, you should say how close delta, how close they have to be. And it's not possible to define it this way. And they have to be small in any case. So the point that I was asserting over here, that the orbit keeps circulating around and around and around. Now because it keeps circulating around that point, it is actually possible to think of defining something like a phase. Because this orbit, if I just project it onto the xy plane, is going around some point. And I can just ask what is the value of tan inverse y by x? Supposing I think of this as some polar representation. And people have actually done that. That you define the phase as y by x, the arc tan of y by x. And then ask how does that phase behave? Because the phase would be normally defined for a cyclical orbit. And for that, if you use some kind of polar coordinates, it's always tan inverse y by x. So just taking that, hence, if I just ignore the amplitude variation in these various cycles, this just begins to look like motion around a circle. I've forgotten the radius over there. And if I'm just looking at motion around a circle, it comes back with a little more algebra to the Kuramoto model. So you should not be surprised to know that if I, if instead of taking n-coupled phase oscillators, I take n-coupled Roslars, I will also find something like the Kuramoto system. I mean, it's not obvious that it happens, but it actually does. And the part of the motivation is the, is the fact that it goes around the circle. Just a second. Yeah? Yeah. Yes? Okay. So the argument really is just by visual inspection, you can see that, just one second, let me get this, that was the last of, it's just the following. See, now you're going around some particular curve over here. Now, it never happens, unlike the Lorentz system. In the Lorentz system, we were going back and forth, you know, going in positive x, negative x, positive x, and so on. Similarly, in y, z was always positive, right? But here, z is, I think, always positive as well. But you'll never find the trajectory crossing, changing its orientation or its direction. So now, if I forget about the distance from the point of circulation, I just have something that is moving around a circle. And so I can replace, you know, I can just say, ignore the radial direction. I'm just looking at the circulation. That gives me a phase. And so, if I have only a phase, then, you know, I think about all the other nice systems which have got only a phase. Kuramoto comes up. Yeah? Okay. Yeah? Okay. It doesn't, see, some of it is a little optics over here, in the sense that this distance of where it has to come before it goes further up is, I'll just try to show you what that is. See, just how far it will jump when is not easily determined by the distance from the illusory fixed point. I mean, the fixed point is not actually in that plane. I'm not sure. I don't know the precise scales. But you can see now how erratically it seems to see the z being large or z being small. I mean, that's where the chaos is. No, but, you know, there is probably some region in the three-dimensional space which is more expansive than others. See, these exponents are not uniform all over. I mean, maybe after the lecture, because I want to get some of these other things out of the way. I want to introduce you to the idea of chaos synchrony. But one can discuss, it's not uniform. You couldn't see that. You saw how suddenly it runs up and goes down. That was the point. That's why you can define a phase. Because if it went up and then came down and circulated over there, then you wouldn't be able to define the phase. It goes up and then goes down and goes round and round and up and down. Yeah, it is an interesting question, though. Is there some proper surface that one can... Maybe we can talk a little more with some more calculations. Yeah. All right, so let me get to... Part of the reason I introduced chaos before coming to the idea of synchrony is that you already know to some level what synchronization is. We saw how oscillators of different frequencies can pull each other. They drag each other. And then as you keep increasing the coupling, they have identical motion after a while. Or they can have anti-phase, which was another manifestation that we saw. But we've seen this both for the case of maps. We've seen it for the case of one-dimensional flows, the Koremoto and so on. The real surprise about synchronization as a paradigm came about when it was discovered experimentally and theoretically, more or less the same time, that you could have synchronization in this system. And this was an observation in the 70s and 80s. All right. Now, you've seen this picture before. This is what the Lorentz attractor looks like. A typical orbit in the Lorentz attractor, it looks like that. And one could imagine that regular oscillators can synchronize quite easily. We had all those examples of circle maps and so on and so forth that it could have anything. But it's most commonly known with the name of Pecora and Karel, but it was already done earlier by Fujisaka and Yamada, and also maybe even earlier by Afraimovich and others and so on. I mean, people have been worrying about not precisely synchronization, but interacting oscillators and so on and so forth. Afraimovich was a Russian who lived in Mexico, I think. Large part of his career was in Mexico. Just saying fact of interest. So what Pecora and Karel did was the following. They said, let me take two Lorentz oscillators. One of them got the variables x, y, z. The others got the variables x prime, y prime and z prime. So I'll make two perfect copies, all the parameters identical. And I know that if I start this with one set of initial conditions and this with another set of initial conditions, they're going to just go wherever they go and so on. We saw the example where I showed you that different points are eventually attracted to the same strange attractor. So different initial conditions should not coincide, and they just evolve on their own. But now what they suggested was in the second of these equations, change x prime to x and wherever x prime appears, just say that instead of taking it from this equation, you're going to take your input from the output of that. So the unprimed system becomes the master, and it gives the signal of x to the slave. So this is what is meant by master-slave coupling. It doesn't matter because whatever the x variable does over there just doesn't matter. It also becomes x. So I'm going to sort of show how this is actually, you can put it in the framework of various different things. But when people talk now about master-slave or pecorakarrel, this is what is meant. That some variable or some sets of variables from one system are given unidirectionally to the other system, and that's it. Now, just to show you how beautifully it works. I think this variable is y. And what you see is what I've done is to take the two copies, green and purple, and I make the purple, the master, at some point, t equals 10. See, initially they are independent of each other. They're moving around, and because of sensitivity to initial conditions, the green and the purple are just on the same object but evolving differently. At this point, we mix them, we make one the master, and after that you can see very, very quickly it's just impossible to tell the two apart. The y coordinates becomes identical. The z coordinate also becomes identical. And as a matter of fact, just one second. I don't have that particular curve, but if I look at the distance between z and z prime as a function of time, that just exponentially goes to zero. So the difference between z and z prime just vanishes. The difference between y and y prime here is sort of visually vanishes, and x who cares. But the dynamics is still on the Lorenz attractor because, see, the dynamics of the master is unaffected by whatever is going on in the slave. So the motion is on the Lorenz attractor. It's just that this other attractor has been brought, the x prime, y prime, z prime, they've just been made to coincide. So, right now, the crucial thing that Pecora and Carroll also discovered when they were doing this work was that this system of equations has got three Lyapunov exponents. If I follow this through and go ahead and do it, I'll get three Lyapunov exponents. Effectively, this is now five equations. So there are five Lyapunov exponents. So when they computed the five Lyapunov exponents, what they discovered was that interestingly enough, if you have these five Lyapunov exponents, three of them are the same as the old ones because that dynamics is not changing at all. But then you've got two more. And whether they synchronize or not, depends on the signs of those two extra ones. I've already said that if you have one Lyapunov exponent that is positive, you're going to be chaotic because the distance is going to grow exponentially fast. What goes on when you now couple this in this way and you ask why do they synchronize, there is some other dimension, out of these five dimensions, it's difficult to visualize, but in two of those dimensions, the Lyapunov exponents relevant to those extra dimensions are both negative. If that is so, then you can synchronize. In particular, if you couple them not by making the X signal the master, but if you made Y the master. So that is to say, have these equations, but now with changing this to Y, this again will not matter. And this particular one, make this also Y and X prime Y. So if you make Y the master and in this geometry, it doesn't work because the subsystem, these are called these additional two Lyapunov exponents, are called the subsystem Lyapunov exponents, those are not negative. Totally different. Converged to the chaotic trajectory. Not in Lorentz because Lorentz is globally attracting for all. This is just a phenomenological, part of it was engineering, part of it is there was some idea that you could use this in secure communication because you take a chaotic signal and because it's chaotic, nobody can predict it. You mask a message with this chaotic signal and have it received by somebody and then you tell them, hey, I was using Lorentz with these parameters and eventually they should be able to figure out what's the actual message. Chaos plus message minus chaos is equal to message. It doesn't, I mean it works obviously but it's not good enough. So we haven't, you know, you can't now try to get off of, you know, Facebook knows this. They know what you're thinking anyhow. But okay, to the point, I just want to say that look, this is something which is a control method. All right, because as, you know, just to see how it goes, but I'd like you to see it a little more in the following way. So here is your master and here is the slave that has come but the master and the copy of the master is these up till here where all the variables are only the primed variables. Now consider that I add these two extra terms. So without these two extra terms, here is your slave and the additional terms that have been added in red over here are like what the, you know, that's the coupling. And it's a coupling with a control objective. Namely, there is this entire branch of engineering called control theory where you say that if I want my objective to be that y is equal to y prime and so on, there is a way of trying to do the calculations and figure out. So, but this is the kind of coupling that will effectively it will do the following. It will give you these equations of what Pekora and Karel wrote down as the slave equations. Or you can say that here is the slave system and I'm going to make a coupling, a vector coupling in three variables. The first term is zero. The second term is what are these two things that you see over there. So if I was to try to think about it as I'm going to do in a little while but let me just give you the idea over here. If I look at these master system and let's just call it x dot is equal to f of x. These are three f of little x is just equal to f is equal to fx, fy, fz. I'm just compressing all the notation and here is master system. Here is my system that is eventually going to become the slave. This is just notation. So what I've written over here, this is the first equation over here. X, capital X is just little x, little y, little z. Capital Y is x prime, y prime, z prime and the right hand side is this is fx and this is fy. Now the question that I'm asking in a control method is let's leave this one untouched. What is the coupling and for later use I'm going to be calling this zeta. So here is my coupling and it is a function of both x and y that I've added over here. And this zeta in our particular case over there is all in suggesting that this is to be seen as 0 x minus x prime plus rho z, sorry not plus, multiplied by rho z prime and x minus x prime times y. So this gives you a coupling and if you want to write, you can also write down over here zeta x of x and y, where this is just 0, 0, 0. So what I'd like you to see in these equations is that both the primed and the unprimed variables have both been changed because of the coupling. If you don't couple them, they're just going independently all over the place. You add two coupling terms and what's the coupling term that you want to add for the x variable or for the sort of unprimed variables you don't add any coupling at all. For the primed variables you have specifically this one. Now the other day I think Thomas was asking is there only one or can there be more or let me just pose the question, how unique is zeta y or zeta x? It turns out actually that it's not unique at all. There are essentially an infinite number of different couplings that can bring you down onto the kind of dynamics that you want to see. But just a little more work before that. So I think it's sort of fair to say that one should be... This was like a huge surprise that you could actually synchronize chaotic systems because everyone said sensitivity to initial conditions they should be pushing you off all the time. But then it turned out that it's actually amazingly trivial to get them to synchronize. But still it had its moment. You always look like you're protesting something. No, no, no, it's fine. I teach at a university in India which is known for its protests. Oh, you can, you will. I mean, what I want to show you, I want to sort of work, this is just 1990. When you come to 2020, you will have added all sorts of terms. People have just explored all sorts of... See, these are couplings. These are couplings. So people have... If you can think of a coupling, somebody has surely worked on it. And depending on the kind of system, the couplings are different. The simplest one is like what one observes with two pendulums swinging in phase and so on. So that is one simple coupling which then we just take the first term and the sign is just X1 minus X2, something like that. Believe me, there are reams of paper that have been used in writing this. Okay, I want to talk about another kind of synchrony. See, once it was discovered that you could make chaotic oscillators synchronized in this way, you naturally then went on to all the chaotic systems that you knew. And when you looked at the Rosler system over here, and I've just put in some numbers for the constants in this story, this number Y, sorry, omega over here does actually play the role of the phase frequency. So that will tell you how you're moving around, at least in the XY direction. So what they did was to look at this system with two of these systems with whatever. And the coupling between the two systems is just this capital C times X1 minus X2 or X2 minus X1, the other way around. See, the original equation just has got omega Y minus Z, and the coupled equations has got omega Y minus Z. Have I made a mistake there? Yeah, well, one of these Z's is wrong. Right now I don't remember which. Lorentz I recognize instinctively. And then I've got these terms over here. This is the coupling. So the coupling term over there. And then let me describe to you what's going on over here. What's plotted over here is the Rosler oscillator in the XY coordinates, right, for one of them. And here it is in the XZ coordinates, right, same. So we've already seen that in the little demo. Here are the XY and Z variables as a function of time for a single oscillator. If I just plot, if I write down one of these signals as an amplitude times a phase in the way in which I describe the phase, tan inverse Y by X, okay? The difference between the phases when the coupling is small just keeps increasing. The difference between them, the phases just drift away from one another. As you increase the coupling, the phases now take these sort of long passages of being almost identical, but then you have these things which are nowadays called phase slips, okay? So you have a phase slip, slip, slip, and it just goes up in steps. And as you finally increase C to whatever value this is, you find that the phase difference doesn't change. Finally, they are having an identical phase as a function of time. So the amplitudes themselves are highly uncorrelated, but the phase difference actually goes to zero. And what Pikovsky et al. did was to, first of all, give it a name. So this is called phase synchrony. So if you want to think in terms of the oscillators, it's not as if the two trajectories are identical, but these go up and down at essentially the same time, right? And this is actually a very common kind of synchrony which occurs in nature. So it obviously got a lot of attention. For example, when you look at population dynamics, in many cases you have the fact that the populations are highly correlated, but obviously the amplitudes of the two different populations can't be the same or are frequently not the same. So while they go up and down at the same time, as oscillators, the amplitudes are very different, and therefore the idea of phase synchronization is important. I mean, there are many biological, I think breathing, I don't know, there's something to do with the breathing mechanism and the pulse or some, you know, there are all sorts of phase relations between that. Yeah? I don't know. See, the point is that this coupling is somehow trying to, it's trying to alter the dynamics and it depends on the scales of the problem. So here the omegas, I don't know whether it was taken as one. This is not my calculation, I'm just, you know, reporting from a standard paper. So the coupling will depend on the particular values of these parameters. But these parameters are chosen such that the dynamics is chaotic. In this system actually you can have periodic orbits as you keep changing this number 10. This number 10 is one of the parameters of the game. So when 10 is very small, thinking of 10 as a symbol rather than a number, so you think of that particular, that constant over there is very small, then you have periodic orbits, and once it crosses some value, for example, when it's equal to 10, then you have chaotic orbits. For that, then the range of C may be something else. Meaning that these all depend on a certain amount of analysis that you have to do. That becomes negative. Yeah. You can almost always recognize by looking at the Lyapunov exponent which kind of thing. This system will also, as you keep increasing C, sorry, but as you keep increasing the coupling, you will actually go to complete synchronization also. So phase synchronization is sometimes just called a weaker form of synchronization. Yeah. I'm sorry I didn't get to you. Absolutely they go. So just a little blow-up of the two trajectories. Here is a case when they're not coupled, when 10 is small or zero, when that parameter, when C is zero, the two of them are just totally uncorrelated. When you have phase synchrony, you can see over here that they are essentially in phase, although the amplitude of oscillation is down. And then finally, when you have complete synchronization, the trajectories just coincide. Over here, what I've done is to give you the same information, but now looking at the coordinates of the two oscillators. So this is x1 versus x2. This is just x1 as a function of time and x2 as a function of time, also drawn one on top of the other. So here this results in an uncorrelated mess. Here the fact that they are in phase gives rise to some sort of interesting curve. It's not a mess, but the phase relation between the two sort of plays out because the amplitudes are not correlated. If the amplitudes were the same, then you get this. And I seem to remember that I was reading some paper which has even that. But then it will always be discovered that it's generally observed that you know synchronization, phase synchronization, amplitude synchronization. But like I say, I don't know every fact most. So here is another plot of systems that are completely uncorrelated with one another. So here is a map, a chaotic map, your standard logistic map where x goes from 0 to 1. And here is the other system also going from 0 to 1. And if there is no correlation, the points are the purple ones just all over the place. Now, on the other hand, if the points are in fact correlated, if they are identical, then they'll all just fall on that particular line. And that line, I mean, it's obvious that they will all fall on that line. But I just want to point out that that line is a subspace of the square. Now, if I take two Lorenzes, and now I'm going to just couple them with epsilon x1 minus y1. So this is my, sorry for jumping between notations because I need to have this notation to do it in a little more. So here, x, y, z has just become x1, y1, x1, x2, x3. And then this is x prime, y prime, z prime, which has now just become y1 by 2, y3. I'm not doing Pecora-Carol, but I'm just adding this term epsilon x1 minus y1 to the first equation. And over here, it will be epsilon x2 minus y2, sorry, y1 minus x1. Just flip it around, okay? I mean, there isn't, you know, the coupling doesn't matter in precise form. I'm just saying that I'm coupling the two and I want to look at what the dynamics does. Okay. When they are uncoupled, you get the purple blob. I've just plotted x3 versus y3, right? x3 versus y3. And I'm just looking, they're completely uncorrelated with one another and I just get that mess over there. Okay, it's not uniformly covering that space, but given enough time, it will probably get there, yeah? Okay. Once I couple them and, okay, if I couple them for small values of epsilon, they are still not going to be identical. In fact, that's the orange mess over here. Okay, so I've coupled them, but the motion is not synchronized and you get there. But if I increase the coupling and I finally get synchronized motion, that lies on this thin green line that you see over there, because that is just sort of x3 equals y3, right? I'm sorry, this is the z variable in the primed and unprimed, this thing. So the way to recognize it for me is now just that this is the one which is always positive, right? So what the coupling does, and when you come to synchronization, you again come down onto the straight line in the straight line is x3 versus y3. Similarly, there is another plot you can make for x1 versus y1 and for x2 versus y2. And again, the fact of synchrony will be a straight line, yeah? We will just come down to that in a moment. I just want you all to... See, this is the same fact of orbits looking, they are the same on top of each other and so on. But now I just say that as far as x3, y3 dynamics is concerned, before synchronization, it's all over the place. After synchronization, it's on a line. The same for the other two variables, x1, y1. So I want to just make this statement. The whole dynamics is taking place in a six-dimensional space, x1, y1, x2, y2, x3, y3. It's a six-dimensional space. But after the whole synchronization process is done, we are on a subspace, which is x1 equals y1, x2 equals y2, x3 equals y3, which is just, if you think about it, there are three requirements out of the six spaces. So the eventual dimension of your flow is three. I'm going to come to how we design coupling. You know, I mean, part of... This is... Okay. This is some little bit of what I do. Yeah. Right now I'm just talking identical. Absolutely. You can make all those variations, but it will still be a lower-dimensional submanifold. All right? The point is that the process of synchronization just... You see, I've talked about how it is a method of control, how it is a fact that you can also talk about it as a method of constraint. All right? So it is reducing things down to a lower-dimensional space. Right? And as far as the manifold business is concerned right now, it's just that the synchronization manifold is just the space on which the eventual motion takes place. That space is given by the coordinates x1... You know, by the condition x1 equals... Sorry, this was... x3 equals y3 times x2 equals y2 times x1 equals y1. Yeah? Don't ask me what the shape of that manifold is, because I mean, we are looking at a subspace of six-dimensional space. I'm already lost at three. But I mean, what I want you to get is the idea that synchronization means confinement to a lower-dimensional subspace. Yeah? Okay. So if one asks the question generally, how do systems synchronize, then the large part of all these says that systems can be mutually coupled to one another, as we've seen in this last example. The term x1 minus y1 adds to all... adds to both the systems. Right? Or as we saw with Pekora and Karel, you've got the master driving the slave. And there's a third interesting paradigm, which is that both these systems can be controlled by one boss. All right? And when you have this condition of both systems being controlled by a third system, this is actually quite... I mean, it's the one that we are most familiar with, namely the circadian. All organisms on this planet have that... get tuned to the circadian rhythm. Right? And we take... All of us take it directly from that satellite there, not satellite, after Copernicus, but... Yeah. But you get the idea, right? Everybody starts looking at their cell phone. That's again coupling to some real satellite up there. Okay. So now I've shown you also how you can have perfect synchronization in many different ways. We can also have variations of synchronization, which is this business of exact synchronization or phase synchronization. So there is a framework for looking at all of them together. It's called generalized synchronization. Surprise, surprise. Right? Okay. So this generalized synchronization, I'm not sure what this is, but anyway, this is complete synchronization, phase synchronization. I don't know why again. And I should change that all to almost all other forms of synchrony that have been seen. People talk in terms of lag synchrony. Because things are exactly the same, but at a small difference in time. Okay. There's also... Oh, there's this relay synchrony. You name it, there's probably a version of synchronization. Okay. So I just want to point out that force systems are a very fruitful way of thinking in terms of different types of synchrony. So if you think of a forced oscillator, for example, okay, exercise in elementary mechanics. So here is an oscillator with frequency omega naught. And if I force it with another oscillator of frequency omega, you know what solutions are obtained. These solutions are, depending on whether omega is equal to omega naught or not, then you just find behavior of one kind. You have beats and so on and so forth. Now, I just want to point out that f cosine omega t can itself be seen as the... It can be seen as the solution of this equation. This equation for a specified initial condition, right? This equation equation over here is another oscillator equation. And this solution is just a cosine omega t, right? So if I were to go back to this equation over here, look at f cosine omega t, right? a is equal to f, right? This is just q double dot plus what omega naught squared q equals y. That is, I've got two equations of motion. One is independent of the other. This is just the mathematics called skew product, right? Or you can think in terms of master, well, not quite master and slave, but this is your drive which is not affected by the response. This is a drive response system. So you have a drive and a response, okay? But now both these systems are not the same. I can't say why is the master, because the way in which I think of master and slave is that democracy, master and slave without the coupling are identical or very close to identical, right? So here I'm not... I couldn't care what omega naught is. I could even have a nonlinear oscillator being forced. I mean, this methodology would still go through, but there'd be just two different systems. One is the drive, the other is the response, yeah? Okay, so the idea here is that we have for a long time been studying these drive response systems as interesting examples of dynamical systems, right? Okay, now when systems are non... Now the thing is to go back and what I want to think of is the following. I've got my drive system which is giving some oscillations. Here it's a harmonic oscillator, but it could be any old... any equation that has some kind of oscillatory solutions. And I take that and put it back. Here again it's driving a harmonic oscillator, but it could be any other equation and it would be the output from here driving that. In the extreme example, one can think of having, let's say, one Lorentz oscillator. That's giving rise to some oscillations which give you some kind of chaotic forcing. And that could be fed into, let's say, the Rosler or another Lorentz or whatever, you know what I mean? Because the interaction between different systems is very important, right? Okay, now then what does one get? Okay, so this is the general methodology that has been applied. I mean what I was suggesting was just this, that I have some equations which oscillate and you notice that this equation doesn't involve Q. Here I've got another equation which has its own dynamics and then it gets affected by Y. We'll see other ways of writing such equations momentarily. Okay, so the point that I've already made is that eventually when we see some kind of resulting dynamics occurring on a lower-dimensional invariant sub-manifold of M of the phase space which is the synchronization manifold, all this means is that you come on to this lower-dimensional subspace and then you stay there. If your synchronization is stable, if your subsystem Lyapunov exponents are negative, then once you have come on to that sub-manifold, you're going to just stay on that sub-manifold. There's another way of thinking about it which says that if I've got these two systems, X and Y which are coupled to one another, again with an abuse of notation, X and Y refer to the variables of these two other systems. So it could be X1, X2, X3, Y1, Y2, Y3. When this happens, the implication is that Y is uniquely determined by X. So I have Y which is some functional of X. So this is the paradigm of generalized synchronization that two systems could be dissimilar. You couple them. It's not perfect synchronization, but if it's going on to a lower-dimensional sub-manifold over there, it's possible to think of one of the systems as being uniquely determined by the other. So the one way in which you would think about it is the following. If you have some drive which is affecting the response, let's just consider this case that you have a drive which you then apply on to the response and you get something that looks like a complete mess, meaning you've got some chaotic drive or an aperiodic drive, it doesn't matter. You've got a response, whatever it is, and you look at what is the output. Pecora-Carol, the drive was identical to the response, and so if you had a chaotic drive over here, your response was identical. It was on that diagonal line. So the way in which people discovered and then discussed generalized synchronization in the 90s was to say that let me make another copy of the response system. Because the response systems intrinsically are chaotic, they are evolving any which way. But after the drive has been applied to this response system and comes up with one output like that, when it applies on to the copy of the response system, it should come out with the same function because of that functional relationship, which means that the two responses should be in perfect synchrony. Drive hits response one, you get some output. It hits response two, you get the same output. So this is the basic idea of generalized synchrony and let me just show you how that works in a very simple system. Now this is the logistic map which I presume is boringly familiar to everybody. So for the case of the logistic map, I'm taking Mr. X as the drive and the response is Y and this function is this little function over here because even I got tired of just saying 4X into 1-X. This is 4 square root of X into 1-square root of X. It doesn't matter. So your response is the same and here is the additional driving term. So when epsilon is equal to zero, these two are just completely uncorrelated. Well, not completely because things tend to concentrate at various points but it fills up almost this entire space over there. This is X on this axis and Y on that axis. If I take another copy of Y which I have now indicated by Y prime, I also get if I look at Y versus Y prime, see X versus Y or Y prime is going to look something like this. Y versus Y prime is uncorrelated. So it's just filling up that space. For increasing this coupling epsilon over here, let me just describe what happens to X and Y. X and Y go from being this space to this. Some complicated thing which I don't know. This is not a simple function but then I increase the coupling even more and then lo and behold, I'm on that synchronization manifold. X is equal to Y. But now let's look at Y versus Y prime. When this value of the coupling has been reached, Y versus Y prime is identical. So X versus Y is this mess. X versus Y prime is the same mess. So the two messes are on this synchronization sub-manifold over here. Increase the coupling more. X is exactly equal to Y. X is exactly equal to Y prime. Y prime is equal to X prime. Everything is equal on this slide. So it's this intermediate picture that will give you an idea of what is generalized synchronization. Response is complicated. We don't know what it is. But any two systems will have the same identical response because there is a unique functional. I mean implicit. I don't know what this functional is because I mean who can tell what gives that. So one drive, many responses. There is a regime when all the responses give you the same output and that is in a sense in the next class. We'll discuss why it's called generalized synchronization. Okay. I'll give you a preview of what is to come. Okay. Just some fact, let me mention. This coupling actually turns out also to replace the coupling by noise. So you have noise acting on one system, noise acting on another copy of the same system. If the noise is identical, the response will become identical. Okay. This has got important applications, especially in things like ecology and so on and so forth. Something called the Moran effect and so on. But what I'm saying is that this is actually a very general framework. If I think of two systems, here they are subject to similar coupling. But if I make them subject to noise or to periodic oscillations or to quasi-periodic oscillations or whatever, the responses becomes identical after a while. Yeah. Exactly identical. These are usually called common noise. So if you think of two nonlinear systems in a common bath, heat bath or something, they're all given the same fluctuations. Not the same spectrum of fluctuations, the same. Okay. All right. So this is it. But usually for most couplings, it's not very long. It just goes quickly. So this is the point that I just made, that the so-called external drive can be even a stochastic drive in many applications in, you know, at least current kinds of applications and complexity. This is actually an important thing. Okay. What I want to do is to now go back and say that if synchronization is sort of bringing everything onto a common sub-manifold, I choose my manifold. Does it always have to be either this line which is diagonal or we saw some example of anti-diagonal? Maybe I can have my own sub-manifold of choice. Will I still call it synchronization? And how do I make sure that it happens? So I want to now think of all of synchronization as basically going onto a sub-manifold in your phase space. Okay. One day. Now we take, we have a group photo. So please.