Hi. In the previous three lectures we talked about tipping points. And we saw two models, right? We saw percolation model and we saw diffusion model. And both of those created tips. What I wanna do in this lecture is give some formal definitions of tipping points. I'm gonna make two distinctions. I'm gonna distinguish between what I call direct tipping points where when I change that variable it causes that variable to tip versus contextual tipping points where the change in some action or the change in some parameter causes the system to change to result in the tip. So in the, in the percolation model in the forest floor model we had a change. A change in the density of trees or in the density of the soil, and that caused the system to tip. So, that's a, a contextual tip. I'm also gonna talk about tips between and within class, right? Remember we talked about equilibrium systems, and complex systems, and periodic systems. Systems can tip from one of those states to another. And to understand all this, I've got to start with something else. I'm going to start with just some very basic graphs describing some dynamical systems, so we can better understand how these different pinpoints operate. So right now, we're not doing the mathematics of an dynamical system, we're just going to draw some simple graphs and that will help make sense of a lot of this. Here's the basic idea of an dynamical system. What you've got on this axis is, here's this variable x, and what this is telling you, this little x dot is telling you how x is gonna change, this is what this line tells us. So if the line is positive, if it's positive it means that x is gonna move in this direction, and if the value is negative, like it is over here, it means x is gonna move in this direction. So if you look at this particular graph, this, of a dynamical system, it shows a stable equilibrium. If I start here, I'm gonna move in this direction, to what, right there. This point y/2. And if I start here and move in this direction, I get here. So, if you perturb the system in any way, we're gonna go right back to that point, so it's stable. Here's a more complicated dynamical system. In this dynamical system if I'm in this region, the value of the function is negative, right, so I'm going to move in this direction. If I'm in this region, the value of the function is positive so I'll move in that direction. And in this region the value of the function is negative, so I'm going to move in that direction. So what that means is if I start here I'll go to this equilibrium, right, and if I start here, I'll go here, and if I start here, I'll go there. So what you get is you get a stable equilibrium here, and a stable equilibrium here. Now notice, this point right here where the value's zero is an equilibrium, but it's a tipping point. Because it's unstable. Any slight change in the variable: if I just move it a tiny bit to the right, I'm going to head over to this equilibrium. And if I put it a tiny bit to the left, I'll head over to zero. So you go from a stable equilibrium to an unstable equilibrium, to a stable equilibrium. This unstable equilibrium, we can think of as a tipping point, because any slight movement, any change in that variable, will lead to a large change in the variable. That's what I'm gonna call a direct tip. So you're sitting at some unstable point right here. And if I just moved it in either direction, right? I'm gonna roll down the hill. I need to go to this equilibrium or that equilibrium. So that's a direct tip. A tiny change in the variable itself leads to a big change eventually, right, in the value of that variable. So, formal definition: Direct tip: small action or event has a large effect on that end state. Right? So you change the variable a little bit, and it has a huge effect on what happens in the long run. So, for example, World War two[sic], right? The killing of Archduke Ferdinand, right? You could argue that this resulted in this tip where the whole world goes to war, and tens of millions of people die, right? But they sort of just tip the whole system, right, and led to, you know, alliances forming and nations raging war against one another. All because of this one act. Right, it was a direct tip. Now, in the Vietnam War, right, one of the things that escalated the war was the bombing by the North Vietnamese of Pleiku. Now McGeorge Bundy, who was the national security adviser at the time, he famously said, "Pleikus are like streetcars." What he meant by that is that streetcars come down the road all the time, that "Pleikus", these events that can escalate war, they happen all the time. So what he was saying, basically, is that even though Pleiku was what I'm going to call a direct tip, right?, that-- Yes, true, it was a tipping point in the war, but they come along all the time. Because what he was saying is, the context, right, the environment was such that it was bound to happen. So if you think back to our percolation model, what McGeorge Bundy was saying is, look, it was gonna percolate. The system was gonna tip. The probability was about 59.27%. So all we needed was one drop of water, and it was gonna race its way through. All we needed was one match, and the whole forest was gonna catch on fire. And that those matches, "Pleikus", are like streetcars. So. The point here and it's an interesting one, is even though we often focus on the direct tip -- we focus on the killing of the Archduke, we focus on Pleiku -- what's really going on in a lot of these cases, that what's causing those direct tips is a change in the context. Right? So let me explain how that can work. So here's our function right, and we've got, you know, this system where, here's a stable equilibrium, here's an unstable equilibrium, here's a stable equilibrium. Well, what happens if the context changes and moves this line down a little bit? Right? Well, if it moves this line down a little bit, then all the arrows point in this direction. So this equilibrium that was stable, right, is no longer stable. So what we've got is we've got a contextual tip. A change in the environment, just by a tiny bit, has an effect such that it wipes out, right, in this case, by the environment causing this curve to shift down, it wipes out this equilibrium. The dynamical system always points to the left. And what you get is you get a completely new equilibrium at zero, right? So, a contextual tip is a change in the environment. So in a percolation model it's sort of filling in more squares. In a forest fire model it's growing more trees. In a social network model it's making more connections. And it's, in doing that it makes the possibility for a direct tip go way up, and it means that the end state of the system is likely to change. So, remember we saw in the percolation model, this is a change in context. It's a change in the number of squares filled in. Remember, also, our SIS model, right? R0 was the basic reproduction number. When we change the virulence of the disease, or we change the rate at which people contact, or we change the rate at which people sort of get, recover from the disease: that's changing the context in which that disease spreads. And it's changing the likelihood, then, if, you know, once the disease gets kick-started, that it's gonna spread throughout the population. So changing the context is what makes these direct tips possible. Okay, so we've talked about direct tips, where the variable moves a little bit, and causes itself to move even further, so that it bootstraps itself, and then contextual tips for the environment changes. I want to make one more simple distinction, remember we talked about four different types of systems: equilibrium systems, periodic systems, random systems and complex systems. A system can tip within a class, so it can go from one equilibrium to another; or it can tip between classes, it can go from equilibrium to periodic, and from periodic to complex. So, for example if you look at pictures of chaos, this is a logistic map, you can have a system that starts out in an equilibrium and then moves to a simple periodic system where it goes back and forth. And then you can get in here where gets these sort of really, you know, sort of interesting patterns that might be classified as complex or even random. Right, so the system can move from equilibrium to periodic to complex as you change the parameters. These are the contextual tips that are between class, right. So. Tipping points, what types are there, right? So there's direct tips, we changed the variable, and that caused the variable to tip. Right? There are contextual tips where the environment changes. Once the environment changes, then the system is likely to move from one state to another once somebody lights the match. There's tips within class, where you move from one equilibrium to a new equilibrium. And there's tips between class, where a sy-, where a system tips from an equilibrium to, you know, a much more complex state, or a periodic state. So that's a simple taxonomy of tipping points. And we've seen models that generate, you know, pretty much each one of those types. And we wanna go next, we're gonna talk about how do we measure tips? So is there a way to measure how tippy a system is? Thank you.
