So, why does not everything then go into chaos? Why if there are so many things happening and constantly interacting into each other, why is it not just dispersing into a total chaos? Again, using the following, the Second Law of Dynamics, why do we have the patterns that you see there in the experiment that you've seen there? So why is it not dispersing into chaos and where does this complexity actually happen? So, for that we need to understand the difference between chaos and complexity, and there's a very simple difference that I will explain to you right now. So here we have a double pendulum, so this is the little pole, there are two moving parts on it, and let's just run it. And when you let go, it will just move these kind of very erratic movements. It's beautiful. I don't know if you've ever done this at high school but it's tremendously cool, I think. It's just unpredictable how it's moving. There's no engine in there. There's nothing driving it. Well, there's gravity but there's nothing driving it. So, this is to some extent a chaotic and what we call the Deterministic Chaotic System. And a deterministic chaotic in a sense that basically, the different patterns that it will follow is different patterns that it will generate which is actually simulated in this part here. This is a simulation of the same thing, a computer simulation. Different patterns that it's following, what is it I want to say there? So it is unpredictable to some extent and the different patterns are the consequence of small differences in the initial value. So, if you start with a certain position and if you just, if you're one micrometer apart from what you started previously, you get a total different outcome, and that's what we mean with these, the deterministic chaos. And, of course, there's this classical example that you've heard about the butterfly in Brazil, I think it was flapping its wings will have an effect on the climate in Europe, or the winds coming in Europe or whatever. So it's basically expressing that a very, very minute changes happening somewhere in such a complex system can have huge effects later on somewhere else. This shows these chaotic aspects and this non-linearity that we'll find in those systems. Now, in complex systems, this kind of uncertainty so deterministic systems, chaotic systems, uncertainty arises from this inability to be exactly started at the exact same initial conditions but in complex systems, actually inherent in the system. It's inherent in the characteristic of the complexity. So it's not because of the initial conditions, that's why it actually differs a lot. Now, I cannot talk about the introduction of complex systems without this slide. This is a world famous slide. It gives a good feeling about things. It's not totally right but it doesn't matter. Like I said, complexity science is still an evolving science. We're still trying to make out what we're actually talking about. So it's not all constant either, not at all. But this is a slide that kind of gives the feeling of where complexity happens. So, it's obviously not in the total chaotic domain. It's not something which is only chaos, because if I switch on my television and I take out the antenna and get this wide noise, well, that's not complex, that's just wide noise. So that's kind of chaos we're looking at here. It's also not totally ordered. If it's just sitting there and not doing anything, not pretty complex, is it? So it's somewhere in between. So it is somewhere in between. This is like a real hand-waving argument, I realized that. You can quantify it a bit, there's been a lot of the work done actually in Santa Fe looking at control parameters of how you move from order to disorder systems, and then, go through these complexity field, fields with complex systems are happening where everybody in this field is still working on that kind of question. Can we simplify it in that way that we just have one parameter to tune and to move from one domain to the other domain? For some cases, we can hold other cases we cannot. What we do observe however, remember that complex systems, these are agents that are interacting with each other in a nonlinear way. What we can do, however, is to look at those systems and try to find some rules, try to find some characteristics. And, one of the things that you can do is that is shown over here, this is from a paper in 2010 by Osorio in Physical Review E., and he was looking at brain cells. And, so he looked at avalanches of neurons in the brain. And, one of the ways he described it and this is literally stepping out of that particular experiment and that's why I like it, so he says, "If the heterogeneity of the agents that we are talking about, if that increases, or if we end, or we have the interaction strength between, in this case the neurons, if that is increasing, then we can have this whole face base of things happening." So, if things are interacting with each other and have a very strong interaction with each other, and they have a low heterogeneity, so they are very relatively simple uniform elements that are interacting with each other if you like, then I get the total synchronized behavior. Everything goes in kind of a lockstep. Everybody is connected to everybody. It's not much freedom to play around with. On the other hand, if I have an increased heterogeneity and not that much coupling, I get kind of almost incoherent chaotic movements. So, it's again the same thing. It's again the same thing like the previous thing where it shows you where you move by choosing a parameter, you move from localized structures through complex structures to chaotic structures. Actually, here you say you see the same thing but then expressed as a parameter in terms of these heterogeneity and the coupling strength of the agents that are there. And it actually allows us a little bit to try to define this kind of heterogeneity, try to define this coupling, and see where we are, and if you can observe different kinds of dynamics in those different regimes that you have here, so the incoherent regime organized itself as criticality regime which is actually a topic also of the lectures, and the criticality regime and then, of course, the more simple synchronized regime. Now, looking at this, you also start to realize that we talked about how things are connected, right? And nowadays, one of the tools to study complex systems is actually look at that connectivity. And you see, I hope that I'm trying to convince you that Complexity Science is not a ready science. It is evolving as we are speaking. It is evolving. We are trying to come up with new ways to describe those phenomena and, I'm just giving you a flavor of the kind of tools that we are developing to study them.