 So, are you tired? No, okay, very good. So, okay, so let's start. Thank you and namaste. I know it's a bit of a cop out, but I didn't have a chance to. Okay, all right, so far what we've done is to look at, you know, some of these phenomena for individual oscillators, also for some collective behavior. But what I want to talk about today is a bunch of topics which have to do with what happens when you have an ensemble of these oscillators interacting as we've seen them, all right. So, but first let me introduce you to this Stuart Landau system. Just a second, and I get this part done. Consider the following coupled equations. x dot is equal to mu x minus omega y minus beta times x squared plus y squared times x. And this is the velocity in the x direction. And y dot is very similar except some signs are changed. So, when you look at this equation, you can immediately see that 0, 0 is a fixed point. And this fixed point, if you linearize around it, you have to take the derivative of this with respect to x, and all you get is mu at 0, that is. If you take the derivative of this around y, you get just minus omega. Basically, once you linearize, you get the following linear equations. And the eigenvalues of this matrix mu minus omega mu plus, sorry, omega plus mu is mu plus or minus omega. So, if mu is negative, we know that that is an inward spiral. And if mu is positive, we know that it is a outward spiral. So, it is stable for mu less than 0 and unstable for mu greater than 0. Now, in polar coordinates, you find that x, in polar coordinates where x is r cosine theta and r sine theta, this just nicely boils down to the following two equations. Theta dot is equal to 1 and r dot is r into mu minus b r cubed. This is your radial dynamics equation. Now, for this equation, 0 is a fixed point because when r is equal to 0, r dot is equal to 0. And mu minus beta r cubed equals 0 gives us that r is the square root of mu over beta, and that's an invariant circle. I mean, it's a fixed point of this equation, but this is in polar coordinates, so it is just an invariant circle. When you rewrite everything, okay, so there are two points to be made over here. When supposing you let mu be negative, then 0 is a fixed point. And as mu changes from negative to positive, or in your orientation from negative to positive, what happens are these eigenvalues, they cross the imaginary axis. When mu is negative, you find that then 0 is a stable attractor. And when mu is positive, it starts spiraling outwards. So if I'm looking at the eigenvalues of this Jacobian, I've got this situation where mu is the real part, that's negative, and this is plus i omega minus i omega. As mu changes from negative to positive, these eigenvalues cross the imaginary axis. What about the dynamics? For mu negative, everything goes down to the origin, 0, 0 is stable, next and why, right? When mu is positive, remember we have this invariant circle, r is equal to square root of mu by beta. So any motion is spiraling outwards, but then it goes and hits the circle, and it can never cross that circle. So this is a limit cycle, right? Because that r is an invariant circle, it can never cross that particular circle. And so we go from a fixed point to a limit cycle. And this is a characteristic of what's called the Hopf bifurcation. This is the limit cycle exists not when mu is equal to 0, when mu is positive. Now, see two things happen when you consider ensembles. If you've got systems that are interacting with one another, the interaction invariably affects the variables of the problem. So when you couple two systems with different kinds of mu's, maybe, and we will see examples of that, when you couple things with different values of mu, suddenly the behavior is not when they are uncoupled. If I took an ensemble of Stuart Landau's, just one second, so finally now if I take the original equations in x and y and change my variables to z is equal to x plus i y, right? If I change my variables to z plus i y, then this is the resulting equation in the complex plane. You've got that equation there, take x plus i y and you can immediately get this equation, yeah? So when I say Stuart Landau, I usually mean just the following equations. z dot is equal to z into mu plus i omega minus beta modulus of z squared. And this equation for mu negative has a fixed point. And for mu positive, it will have a cycle. We just start with those two equations. Don't ask why. When you start with those two equations, you can look at, there's only one fixed point which is zero, zero. You look at its eigenvalues, it is mu plus or minus i omega. And because of the nature of those eigenvalues, if mu is negative, then I know I have a stable spiral, everything goes into the origin. A little analysis tells me that this is also, the representation of the same two equations, yeah? If it's the representation of the same two equations, from here I can figure out that r is equal to zero is a fixed point and r, and then I can, maybe it should have been a square over there, yeah? Okay, sorry about that, it should be the square. So over beta square root is an invariant circle, yeah? Apologies, read this as two. It should have been obvious also because it's x squared plus y squared, r squared, yeah? Anyhow, now the thing about an invariant, anything is that once you come on to the invariant set, you're stuck over there. So you cannot cross that invariant set. So once you have mu positive, what will happen is that any trajectory in the neighborhood of zero will spiral outwards and go and hit that circle and then circulate around it, yeah? So the behavior as a function of mu is that I go spiraling inwards and for mu positive, or let me not draw it in the plane, I get a damped oscillation going to zero. For mu positive, I'll get sustained periodic solutions. So the Stuart Landau is a very nice model for the origin of oscillations and many systems become unstable by fixed points that are stable behavior changing to oscillations and then beyond everything else. So the Hoff bifurcation is a very important bifurcation in all of dynamics and seeing when a system undergoes a Hoff bifurcation it tells you that interesting things are happening. Now if I'm on my invariant circle, that is, I start out over here and I eventually come on to the invariant circle, right? The radius is exactly equal to square root of mu by beta, right? Set mu equals one, beta equals one, and then I get one plus i omega minus modulus of z squared, right? Frequently people like to deal with that. Your radius is one and, you know, things are somewhat simple and your rate at which you're going around, the frequency of oscillation is omega, right? So theta dot is equal to omega is your other equation and lo and behold, we have another way of seeing that the Kuramoto comes out in the system also. So if you have a set of coupled Stuart Landau equations, the dynamics will not be very different from the Kuramoto system. You have to couple them appropriately and all that, okay? Namely, if the amplitude is invariant, if the amplitude is one, right, then all you really have to bother about is the theta dynamics and it's a Kuramoto model, okay? Now that we have, okay, so we now basically have two elementary oscillator systems. One is Mr. K, two are Messers, S and L, Stuart Landau, and a third one which has many of the features of this, we saw it the other day, a chaotic oscillator, the Rosler, right? And that moves around more or less like a phase oscillator. It's not ideal because the amplitude keeps varying, but we'll see that there are many things that it shares, right? So in the remaining two, three lectures that we have, what I want to do is to discuss the different kinds of collective behavior that we find in oscillators of this kind, right? For nothing, what you can do is to take the underlying topology of your space to be something which you might find interesting. So if you take Erdos-Renyi networks or small world networks or scale invariant or whatever kinds of networks, I presume you all have all been introduced to some of these ideas, right? So when I described the Kuramoto model, everything was coupled to everything else. So it was basically an all-to-all coupling. If you thought of it as individual nodes, it was this couple to that couple to that couple to everything. You know, it was basically even. But one can equally well go on to, you know, put stuff on a Bethe lattice, meaning you can play around with the underlying network, right? Where, you know, if these are nodes i and j, a i, j is equal to 1 only when a i and j are connected and is not equal to 1 when they are not connected and so on. All that can be done. The kind of collective behavior I want to talk about, let me just tell you what I, I'll talk about some or all of these, all right? Supposing you put oscillators on a certain kind of network, what kind of synchronization do you see, right? One that I will look at are these so-called bipartite networks. So you look at networks of this kind where networks in partition A are not coupled to other networks in partition A, they're only connected to partition B, right? So I've got these six nodes, I split them into two sets and everything in A is connected to things in B but not to anything else in A. This comes up a lot in food webs or various kinds of relationship networks or I don't know, country A, country B at war with country B, but not with country A, for example, all right? Taking no names of country A or B because it's going to fly anywhere, all right? Okay, and like we have, this is two partitions, you can have n partitions, k partitions, you can do, and then you can ask what are the stable patterns of oscillation? There's a very interesting phenomenon that happens and it happens with Stuart Landau's that if I take one Stuart Landau and another and I couple them, right? If I couple two Stuart Landau's, then what happens is that these, depending on the strength of the interaction, these two oscillators can drive each other into lack of oscillation, right? So effective, see, remember that lack of oscillation over here meant that mu was negative. So even if mu is actually positive, if I couple them strong enough, they drive each other into a regime where mu is effectively negative. So this is a phenomenon of what's called amplitude or oscillation death and this is an extremely important phenomenon that occurs and that is something that we experience quite often. When you sit in a car and you know that there are springs and what not, you know that keep your suspension going. Now if you just allow your suspension to keep going, you're not going to have a very comfortable ride. So you need to be able to damp out this kind of oscillation and this damping of oscillation happens often by making the oscillations die out or oscillation death. I'll show you some examples of that. An unusual phenomenon that not on this chair, I mean you know you have those friends who are poor enough and can't get their suspension fixed and is going with like that all the time and the little dog in the sitting in the back has got his head moving around. You've seen those cars. An important sort of theoretical question which I'll get around to a little more tomorrow is Chimeras. Chimeras is the following situation that you've got identical systems with identical dynamics, identical coupling, identical everything and then there is symmetry breaking. One group of these oscillators goes into a completely correlated phase while the other goes into an uncorrelated phase. So in the Koromoto system that we saw, what we find is that there is one group which is locked and the other group which is incoherent. And this happens, I don't want to make it mysterious but this seems to happen quite often and it happens for very particular kinds of coupling. The coupling we've seen is either all to all or global coupling and so on, maybe even local, I don't know. But if you have non-local coupling, that is coupling that depends on the distance between these oscillators so to speak, you find that you have this chimeric behavior where some oscillators separate out into coherence and some remain incoherent. And all these phenomena have been illustrated largely with phase oscillators or the Stuart Landau. Now this phenomenon of amplitude death was actually observed chemically. It was observed in an experiment by Barely in Physica D 1985 or so and basically what he's, I mean the title of the paper is oscillators drive each other to fixed points and seize oscillations. So this suppression of oscillation was then modeled by Aronson et al. and what Aronson and company did was to take two Stuart Landau and couple them linearly like this and K is the strength of coupling. Now when K is equal to zero, the two oscillators are uncorrelated, they just, 00 is unstable and the oscillators just go to the limit cycle where the limit cycle would have been dependent on beta and mu, both of which are one so it doesn't matter. So the oscillators tend to stable limit cycle states z i equals one and when K was not equal to zero, they allowed K to go from zero to three and delta, delta is the difference in the frequencies omega one minus omega two and they let that go from some number to some number and they observed the following. This is the level of observation. For small delta and as you increase K, you have a large phase of phase locking. In this phase diagram, you have a large phase of phase locking. If you keep the coupling small and allow the difference in frequencies to go high, you go to a drifting state. So I mean if you remember the Kuramoto, which is again very similar to this, if the coupling is strong and the difference in the initial frequencies is small, then we know that the phases must get locked as a function of time. If on the other hand the coupling is weak and the difference in the original frequencies is strong, then is large, then two of them are more or less independent, they are moving around the circle on their own. What Aronson et al. did was to find that if you had intermediate coupling and now it didn't matter on the difference between the original frequencies, you go to this stage over here, a large stage of amplitude death, namely everything just comes crashing to a halt. So I mean this is unusual because you've got two oscillators, you just couple them and boom everything vanishes, all the motion, all the dynamics goes away. Now notice over here that time is implicit, Z1 is a function of time, Z2 is a function of time, and the interaction over here is K of Z1 at a point in time minus Z2 at the same point in time. Now if I think of this as system 1 and this as system 2 and I'm thinking of interaction between them, then you know why should the interaction between them occur at the same time because after all some signal has to go from here to here or vice versa, right? I mean everything is not happening in a vacuum and it's not necessarily happening simultaneously. So the first innovation that Jonson, Sen and company did was to say that take the second of those two equations, the signal from Z1 at time t minus tau that reaches this second oscillator only at time t. So the distance between these, some velocity, some impedance, you call it what you will, okay? But this is a very important kind of innovation in generally in differential equations and this is to introduce time delay. Time delay has been historically very, it's both necessary to include and difficult. It's difficult because in order to know this trajectory at time t, I need to know the entire history all the way from t to t minus tau and as I advance more, I need to know more of the history and since I already know the history from t minus tau to t, then when I go to 2t, I've got to still keep a memory of everything that's been happening, okay? So this is not, these are not easy. Have people any experience in solving time delay equations? See, no. See, this is not a, this is not like standard stuff, although it should be. But I just want you to appreciate that time delay means now actually increasing the dimension because I need to know the history, detailed history all along the time. So what is a one, you know, this is now a one freedom equation. Okay, it's complex, but even if it's four equations, it actually becomes four times infinity. The moment you introduce time delay, your equations become infinite dimensional because you need to know the entire history at all times, right? In the discrete case, it is a little less because there, if you depend on the step at n before, then your dimension only goes up till n, but for continuous systems, it goes to infinity, right? And integrating equations of motion which have got time delay in it is not the most difficult thing in the world. I mean, you discretize infinity 100, 200, you know, something like that. You always take some approximation for infinity, okay? So anyhow, when you do that, okay? So the reason for this, from a physical point of view, the reason is signals transmit with finite time typically. And the other time, the other reason by, I mean, what is bringing the signal from here to here may actually be a diffusive process of some kind. So the time delay is actually, you know, what you're modeling over there is diffusion, right? Okay. So now you take two time delay coupled Stuart Lorentz's, right? And you've just done that in this term, right? Okay, so Z, you just put a time delay in the opposing variable. So Z1, you put a time delay in Z2, and for Z2, you put a time delay in Z1, okay? Because this is the communication delay, if you like. And Sonja, all right. Now you see what happens to the phase, okay, oops. So this was the phase diagram for the Stuart Landau where you had, okay, I just want to concentrate on two things. For zero delta, that is even for identical oscillators, right? For identical oscillators, also your amplitude depth region started there regardless of the coupling. And your phase locking was instantaneous if your frequency difference was small for all k. What happens now when you put in time delay is that this phase diagram changes significantly. And most importantly, you get a region of amplitude depth even when the frequencies are identical, as soon as I increase the coupling, okay? So there is some kind of interactive feedback. I forget what, oh yeah, there's the tau. The tau is not very, you know, these are all some dimensionless units, but you know, the one and nine, ten, things like that, you know what I mean? So for fairly small time delay, automatically what happens is that region which you saw over there, which is like a triangle that's come all the way down to this axis and you've got amplitude depth, okay? So yeah, I know, I know that this is a situation which is called integrative time delays or distributed time delays been studied, okay? I mean, as I told you in my first lecture, a lot of what I'm talking about is A, other people's work and it's been done over a long time, all right? So this is more to give you a flavor of the kinds of issues that are important. This is not the most current thing. So, you know, people have worried about the point that you're making that why is time delay fixed? So people have taken various models for time delays. The time delay comes out of a distribution. In some situations it's very large, very small. Some people have looked at integrated time delay, that is, you know, the time delay keeps increasing as you're going on. I don't know all sorts of variations have been studied, right? And now I've gotten to the point where I start forgetting what I have done. Okay, but look, this is an important figure. To tell you that two identical systems when coupled with time delay can actually make both of them stop. Okay, so now this is an effect, as I said, has been extensively studied and it has important applications in stabilization and control of fluctuations, namely that. It's been, it was originally thought to be possible only in limited, you know, because it was first seen in time delay or it was first seen in coupled Stuart Landau so they said, okay, it's a peculiarity. But since then people have worried about it and seen that it occurs everywhere. Now, what I'm showing, okay, no chaos here. I mean, take two roslars and couple them appropriately, you'll make them die. Yeah, yeah. Okay, the phase locking over here is synchronization. So in this, what I'm showing you over here is a picture of the largest Lyapunov exponent in the system. This is a four-dimensional system, right? Because there are two equations for this and two equations for this. They're separated into real and imaginary parts, right? Once I do that, if I calculate the largest Lyapunov exponent that's what it, the largest two look like this, right? This is the region of amplitude depth. Let me just, oops. What I'm doing is to go across some line like delta equals 2. I mean, if you just think about it, what does amplitude depth mean? Everything goes to a stop. So what are the Lyapunov exponents when everything has gone to a stop? It's negative, right? Okay, here you have dynamics. It's positive, but it's a non-chaotic system so the maximum you can expect is zero, right? So as I go across this particular line, I expect zero Lyapunov exponent negative zero, right? And that's to show you what that is, right? So I'm sorry I don't remember the actual number of, you know, what is the value of delta over here, but as I keep changing tau, this is, okay, it's the third dimension of that picture, right? That was K delta and I can now put tau as another axis and ask what happens. So as I keep increasing the time delay, at some point they go to the depth area and the depth area is always something that looks like so because as I keep increasing the time delay further, I go out of this condition that will give me death. See the phenomenon of amplitude death is not that once you have gone to die, that's it forever, there is also resurrection, right? So that's amplitude resurrection if you like. And I mean a lot is understood about diagrams like this. It turns out that there are islands of amplitude death and these islands of amplitude death are surrounded by oscillatory behavior. So what you find is that and in the region of death, all the oscillations are damping out. So the damping out that you saw is basically this, that you had something that was oscillating and you increase the time delay, boom, it started spiraling in words and it went into the fixed point. In the previous picture K was fixed. I think the purpose of that illustration and that I think is my work was to show that, look, it was 10, you know, 20 years ago. The purpose is to show that K is a control parameter, tau is a control, so for a given tau, it could happen that you will find amplitude death over a certain range of K. For another value of tau, it will be another range. So what you have are these, if you look at K versus tau, let's say, you may have an entire region where you have amplitude death and this region can be here. They appear all over the place and where it appears over the place depends on the nonlinearity, depends on, you know, a variety of properties of the system and the people who have been doing a lot of this work and he's done so much in it that you might as well call him saint rogats. No, I mean, he's done a lot of work in this general area of oscillators, oscillator death and so on and so on and so forth. This is blasphemy, especially in Italy. Now, there's a subtle point over here. In the case of the Stuart Landau, you go zooming down to 00, which is a fixed point of the original uncoupled system, okay? Such a phenomenon is called amplitude death. You could, in principle, go to two new fixed points. I mean, you could go to a fixed point which is not the same as 00 and why would one do that? Notice that over here, this term, when Z1 goes to zero and Z2 goes to zero, that term vanishes, okay? And coincidentally, 00 is the fixed point, right? If this term doesn't vanish at 00, I mean, meaning if the coupling is such that, you know, you add some term over here and some term over there, the coupling doesn't vanish over there, then the system will eventually go to some other new fixed points. That in this business is called oscillation death. Amplitude death is just a subset of all oscillation death phenomena. The idea is that whatever was oscillating has stopped oscillating. If it went to the original fixed point, it's called amplitude death. In the other case, it's just called oscillation death. Unimportant, but you will see both terms used. In the business of chaos, et cetera, et cetera, one way to prove that your ideas are right is to implement them on some experimental system. The experimental system they chose was electronic circuits. I know almost nothing about electronic circuits except that they are there. And so what you see is that here are two uncoupled electronic circuits which are oscillating. And when you cross, okay, so you increase some coupling and then they are still oscillating, and then you increase the coupling even more and they come to fixed points. So, I mean, this can be shown with electrochemical circuits with all sorts of things. Now, one of the reasons I brought this up is that from the point of view of electrophysiology, we all have one oscillator within us and whose good operation of sustained oscillations is vital to our lives. So, when something tampers with the oscillation of the heart, for example, can have pretty serious effects. So, it's important to understand this phenomenon of oscillation death or amplitude death, especially because, as I showed you, there is possibilities of oscillation revival. So, many pacemakers, et cetera, they work on some very crude way of doing this. I don't know. I mean, it's a thing to remember. Now, there's a whole range of coupling conditions under which you can find oscillation death. You know, how it is that you find something once and then you're saying, wherever you look, you find the same thing all over again, right? So, there's a paper which looks at external driving that will lead you to oscillation death. In the sense, I mean, that's not so difficult to understand. See, after all, as far as this oscillator is concerned, this is just another oscillator, which is giving you some signal. I mean, we don't care what this equation is, but if Z2 of t is some periodic function of some kind, it shouldn't surprise us too much that that also goes to oscillation death, yeah? So, that's what Konishi did. Then there was another paper on velocity coupling, and then there was a paper on nonlinear coupling, which I will show you because I was loosely involved. And this is on Hinn-Marsh-Rowe's neurons, which I know some of you are looking at, or is it Fitzhugh-Nagumo? You're looking at Hinn-Marsh-Rowe's. Fitzhugh-Nagumo. All of a family, right? So, what we did was to look at these equations for X dot are just the Hinn-Marsh-Rowe's equations. And H is a coupling, which is got time delay in it, right? And so, that's your Hinn-Marsh-Rowe's equations there. That's your coupling equations. And this coupling is not X1 minus X2. This is a nonlinear coupling, which comes out of phenomenology. This is work done with a neuroscientist who said that the correct coupling has got some voltage in it and it's got some, whatever, some functions. And here, the signal from the jth neuron comes in only after a certain time talk. So, if you take this system, this also goes to amplitude depth as a function of the coupling. This is the largest Lyapunov exponent. Below a certain value, that's what the largest Lyapunov exponent is. And after this, it's all non-positive. So, the oscillations have just died out. And, I mean, one of the observations we made was that time delay amplifies the region of amplitude depth in these systems, right? So, back to the Stuart Landau. I just want to point out some of the sort of, even the region of amplitude depth is not without any interest. See, in one case, you have the largest Lyapunov exponent going negative. These are probably chaotic regions, or at least they're, sorry, these are not chaotic regions, but these are just telling you what happens at the fixed point. The fixed point, the eigenvalues are positive, meaning that it is unstable, right? And at this point in the region of amplitude depth, all the fixed points at the, all the eigenvalues at the fixed point become negative. So, you actually have, you know, no zero or positive Lyapunov exponents or eigenvalues at this point in this range over here. But if you look at the dynamics in this part, you have damped oscillations. And if you look at the, if you look at the dynamics in this part, you also have damped oscillations, but there are two of the oscillators, they're out of phase. If you look at the dynamics over here, these are oscillations, because that's where just the simple Stuart Landau behavior is. If you look at the dynamics over here, they're also stable oscillations, but they are out of phase. So, the region of amplitude depth contains within it some place where the flip, the phases flip from being in phase to being out of phase, right? So, I don't know how significant that is, I mean it is mathematically true when I observed it and so on and so forth. But it's just, just to say that just because all your oscillators are dying over there, it isn't as if the dynamics is just go straight to the fixed point and go straight out of the fixed point. There are interesting dynamics that take place over there, okay? So, so much for amplitude depth. I may come back to that. Tomorrow there are two classes, right? So, depending on my mood. No, I do want to consider, you know, I do want to discuss self-organized criticality and neuronal avalanches and stuff like that also. So, we'll see. Okay. That is my attempt to find an Indian chimera. You know, it's traditional in any talk of chimeras. People will show either, I don't know, usual Greek chimera, right? And I want to let you know that our culture had, we had our own bizarre chimeras in any case. Okay, so the point is that if you have an ensemble, okay, today really we're talking about the qualitative behavior of ensembles, right? So, if you take an ensemble of Stuart Landau's or better still, Kuramoto's and couple it with a non-local coupling, then what happens is that spontaneously these divide into two sets or more. The motion of one set is completely coherent. The motion of the other set is drifting, okay? And there is nothing to specify which oscillator will go into the drifting set, which oscillator will go into the coherence set, or whether they will always remain as members of the set so they could drift in and out. And I'll just show you, yeah, welcome to the gang. Meaning that, okay, this is, you can see it as an effect, you can see the mathematics and see everything, but you know the reason for the spontaneous symmetry breaking is not, I would say the jury is, if not completely out, at least there are parts of the story which are not well understood. I will discuss this, okay? Like I said, today I just want to talk about the phenomena that are occurring, and then I'll come to it, okay? Now, this was discovered experimentally in some sense by, first it was discovered by Kaneko, and then it was discovered by Kuramoto and one of his post-docs, Mongolian called Batog Tok, and then it was observed by Strogats who was baptized as Chimeras, right? In the sense that he said, this mixture of stable and unstable behavior together is like an object which has got the head of a lion and the tail of a cow or something like that. So Strogats was responsible for the name Chimeras and it has not surprisingly stayed, right? Now, why do people think this is important? See, various types of mammals and birds experience hemispheric slow wave sleep. Among the mammals are also students in a seminar. You switch off one part of the brain and sleep over there and the other part of the brain is awake. Yes, yes, yes. So you just tell me I'm in my chimeric state, all is forgiven. No, but dolphins are supposed to do that, right? And so on, okay? There are these, there's this phenomenon, a slightly scary phenomenon called ventricular fibrillation and at that time part of the heart is oscillating and part of the heart is not and you have this situation. There are these so-called bump states in neuronal coherence, localized regions of oscillations accompanied by areas of quiescence, all right? And in any large group, there's always people who are out there protesting with their hands up and there are others just standing around and saying, let's see whether he gets killed or not. Okay, no, so this mixture of, sometimes we don't understand the origins of the symmetry breaking but here's a model which can do that. I wish this was a slightly more, okay, all right, this is an experiment done by Capitaniac etc. So you can see here the smallest possible chimera state. The two outer pendula, I don't know which one. See they're all oscillating together and like I said, I wish the experiment was a little clearer but maybe the next one is. Okay, so these ones are exactly out of phase. These ones are exactly in phase and this one is a chimera where you see the three of them on the extreme left. These three are going all happily together. If you just take these three as one set, these nine oscillators are all oscillating in phase. These are oscillating in phase, both of them are oscillating in phase with each other. Here you've got nine and nine oscillating outer phase and these are having their own trip. So this is a very nice paper in PNAS in 2013. Again, you'll get all these notes so you can take a look at them. Spiral chimeras. Okay, so mathematically, these chimeras are complex spatial temporal patterns where coherent and incoherent regions coexist in a system of identical oscillators and it was first reported by Coromotor and BatogTalk, although really it was done by Kaneko earlier. I know that many of you are familiar with the work of Kaneko, a couple map lattices. I know we talked about it yesterday. So when you study large complex systems and ensembles, you find these experimentally. Okay, so the geometry that Coromotor and BatogTalk had looked at was a non-local coupling, that is to say not just the local or the global coupling, which we've seen in the Coromotor study we did, but in this situation where the interaction between one and two is, and one and three is attenuated by the distance, one and four is attenuated by the, and you have a coupling that just goes off with the distance between these two, yeah? So it's not uniform coupling, it's not identical coupling, right? By identical all I meant was that this is just to the nearest neighbors. Here it all depends on the distance between the two, so it's called non-local coupling. Now going again to the continuum description, you now write an integral equation or differential integral equation. This is the phase at position x. So now you're thinking of each oscillator, each phase oscillator actually being at some location x. When we started we just said d phi by dt and d phi sub i. So that was the ith oscillator, but now let me think of x as the spatial location of the oscillator. And it's got its frequency omega, you've got sine of phi x minus phi x prime, so it was like sine of phi i minus phi j, but now I have a kernel g of x x prime, which typically I will be considering as decreasing with x minus x prime, right? So the further away they are the lower is the intensity, but I don't have to, I can consider anything I like, right? So, and there is also an important parameter called alpha over here. This does not just change the phases slightly, it's actually an integral parameter to this model known as Kuramoto-Sakaguchi parameter, right? You will see that. Okay, so for convenience we just take omega as usual from some unimodal distribution, g of omega, and at long times this is what you find, okay? This is the original Kuramoto, I just wanted to remind you, right? What you find is a bunch of phases that are very tightly packed close to one another, a whole bunch of phases that are randomly distributed in minus pi to pi, right? And, okay, this is just the way in which they are drawn, okay? This is out of a thousand oscillators, the way you collect all the ones that are, because there's a way of drawing this picture, but the idea is that there's a whole lot which are coherent, a whole lot that are drifting. And this is a typical picture of a chimera because as per this equation, there's nothing to separate one set of oscillators from another, okay? And so at some point in time I said, let's just look at what kind of non-local coupling one can play with. And I have a fondness for piecewise linear form, so that's my approximation to a unimodal distribution. Oh, it's unimodal. I guess it's non-differentiable at that point, but who cares? Well, the point is that, okay, this is a simple enough kind of form to do. You can do many of your integrals quite easily and analytically and so on and so forth. And it was quite a surprise for us to find that this, so if you took this as your kernel for coupling, which is just triangular, you got a nice chimeric state with one coherent cluster, one incoherent cluster. Just ignore this line because this is some order parameter. If I made it look like that, it turned out actually there were two coherent clusters and two incoherent clusters. If I made it unphysical, that is to say that the coupling increased as the distance went by, I again got, okay, this is a zoology. I mean, part of the game was over here that the number of coherent and incoherent clusters depends on the form of your kernel. And it depends on the form of your kernel in a very specific way, yeah, cluster them. You cluster them into seeing what are all together. See, because this is a meaningless, sorry, it really doesn't matter what your, okay, this is the oscillator index. So the oscillator index, it's not as if oscillator one and oscillator two are going to be necessarily in the same phase, but what you do is you try to sort these out and try to see which are the ones that are all closed by and so on and so forth. Yeah, this is a plot in a way which is most appealing to the point that is being made. There could be, I mean, you're right, there could be one or two over here which actually should be there or vice versa. Well, vice versa, yeah, perhaps some of these should be over here. It's, there is a little art to drawing this, but the fact that they separate out into two groups is not a query, yeah, yeah. All these are phase lot, one oscillator. And all these are phase unlocked or drifting. And the point that your colleague made over there is that could some of these be here and some of these be here. And to first approximation, yes. I think when you're a little more careful, you can figure out whether if you've chosen one particular oscillator to be part of this coherent group, you try to see whether it is in fact coherent or not. So it's not entirely, you know, it's not just entirely whimsical ordering. Once you've done the ordering, you make sure that the client of the coherent group does in fact stay coherent with all the others. I mean, it's a little extra work, but there can still be errors. I'm not saying that they can't be. Yes, Tomaso? Yes. Yes. Yes. The only thing that has changed is the coupling. The coupling has become non-local. See, in the original, this was just for compare and contrast, okay? Over here, I mean, supposing I took the discrete Coromoto model and I made Aij equals one over modulus of A i minus j. That would make it non-local. Which one? The kernel doesn't actually have to be unimodal at all. I mean, I think the kernel has to be symmetric, but other than that, nothing more. It's just symmetric around zero because x can go... Yeah, okay. So there's just more cartoons of what these multi-component chimeras look like. So there is one part which is more or less coherent and the other parts. And you can see actually that they are not adjacent necessarily. The incoherent ones are moving as they like. I'm sorry, by now I have actually forgotten what C1, C2, etc. But typically they refer to the number of coherent parts. Then the final part is that when you have these piecewise linear or even smooth kernels, it is possible to design a specific number of coherent and incoherent regions which are in or out of phase with one another. That means you can do a lot of engineering with this kind of... this kind of kernel design, let's say. All right. Okay. The next point that I want to just allude to is that you can see in the simplest KuroMoto model we've seen this over and whatever before. And now you can ask what happens when... Just a second. I'm not sure what this is hiding. So we've looked at KuroMoto model with... We looked at Stuart Landau with time delay. You could also look at the KuroMoto model with time delay. The simple KuroMoto with time delay. And ask when do you get synchronized solutions? And for the synchronized solution, we just want that the frequency is the same and perhaps there's a phase difference alpha sub i. And you can... I mean, it's not too difficult to show because these are all simple algebraic equations that the frequency of the synchronized solutions must satisfy some kind of a transcendental equation and this transcendental equation will have solutions. One solution is that all oscillators are in phase. Another solution is that the solutions have mixed phases and these can be either randomly distributed or they can be splay states. By splay is meant the fact that the phases are on the circle but they are uniformly distributed on the circle. So if there are two groups, they're just at 90 degrees from... at pi from each other. If there are three, there are, you know, at the vertices of an equilateral triangle. Four square and so on and so forth. Now these splay states are interesting because again these are coming up spontaneously. Right? Now when you take globally coupled oscillators with time delay, you have in the strong coupling regime only in phase synchronized states remain. And for other situations you will find that splay states become, you know, become stable and these splay states are the ones that are marked in zebra stripes for two oscillators, three, four, five, eight, et cetera, et cetera. As you keep increasing, splay has become a little more difficult. All right? Okay, so global coupling plus time delay results in the system taking distinct synchronized solutions depending on the coupling strength. All right? And you'll find the same solutions for Stuart Landau or for the Rossler. Now particularly important kind of network as I already pointed out somewhere over there is the so-called bipartite network. And by bipartite, all I mean is that there are two natural groups of oscillators which are only connected to oscillators of the other kind. So here it is. This is very common in computer networks, for example. You have a hub or a common server and everybody else is connected to the server. So none of the reds are connected to any other red. The blue, there's only one over here, but even if you had more than one, the blue is only connected to the red and so on and so forth. Oscillators on a ring, so long as it's an even number of nodes, that's a good example of a bipartite, right? A linear chain with alternating ones. This is the kind of oscillator that I'm going to look at. The reason we started looking at this and it is a thing to look at is that in the simple case of synchronization, the following geometry is largely explored where you've got three oscillators or whatever, three anything connected in a line and it was observed that oscillator two would synchronize with oscillator three, leaving one alone. Namely, if you had three in a ring, you found one and three were getting synchronized whereas the intermediate one was not. So this was called, for want of a better name, relay synchronization. So the coupling was relayed through the intermediate one. So now you just expand the relay to the star. So the star topology is something that people look at in great detail. This is the simplest kind of bipartite but where everything is connected to everything and these are not connected to blues and that is not connected to red. So when you do the Kuramoto on a bipartite lattice with frequency omega A for partition A and omega B for partition B and all these have got their obvious notation and tau is the time delay. You can do a fair amount of analysis and ask when do we find synchronized solutions and actually this is a pretty enough system that you can prove a few theorems about when you will find in-phase solutions, when you will find anti-phase solutions and so on. I'm not going to go too much into this but here is a situation where you actually have multi-stability. You have many, many solutions becoming possible and there are solutions which are in-phase, out-of-phase and multi-stability regions and so on. I'm not going to... Let me just show you some phenomenology because I don't think this is crucial. When they are out-of-phase, you find exactly the green and the red. This is anti-phase synchrony. This is completely synchronized and there are regions where you have both of these occurring so you have actual multi-stables so it depends on the initial condition as to where you go to. This you can find in the Rossler. There was a question, does it only occur in Stuart Landau? No. Here you find in the Rossler and this is what it means to be anti-phase for a chaotic oscillator or in-phase. When you are in-phase in a chaotic oscillator, the amplitudes are not correlated. You find that they are all going up and down at essentially the same time but the amplitudes are different. That's why you can see the green bits over here. Over here they are anti-phase so this is the meaning of anti-phase or in-phase for chaotic systems. They are not going to be identical but a couple like that. I have some more results for this. Let me actually stop here today because I want to come back tomorrow to talk about some other important parts of synchronization and or collective behavior. In particular, what I am hoping to do tomorrow is to introduce those of you who have not been introduced to self-organized criticality. We are at least one simple model and also indicate some places where it is used in neuronal systems because a lot of memory, plasticity, etc. is associated with the firing of neurons together. One of the more exciting recent developments is to use SOC to try to understand neuronal dynamics. If time permits, we shall also look at creatures that swarm and sink like birds flying south for the winter. I have also given out two distribution sets today. One is some reading material and one is homework. I hope you all will all try to do question one and question two is a gift. Do it, don't do it. I think that there is something interesting over here but it is pretty much up to you.