So what does all this mean? What is the importance of this? What are the lessons to be learned from chaotic dynamics? Well I should say I don't think there is a simple or single answer for this. There isn't uniformed agreement on how important chaos is or what is important about it. And for sure I think there are different lessons or implications for chaos in different area of science and social science and in life. So, I'm not going to have a simple conclusion here. So instead let me make a few more comments and give a couple more ways of thinking things. And I want to start inverting how I've been talking about things so far which is to begin with an equation and see what it can do and of course that is not really how life works we don't get handed equations We get handed life and sometimes we turn life into data. So let's imagine going in the other direction. So suppose we had data. Maybe this is data for population of rabbits on an island. Or who knows it could be anything But I'll talk about the rabbits. And these are 40 years of data. And we can see that the population settles into two year cycle very quickly. and that appears to be stable. And maybe you know some year--the rabbits something happens to the rabbits-- There are a bunch more or a bunch less We would see it returning to this behavior. And we could tell ourselves a very simple story. Of why it is the way it is. Some years when there are alot of rabbits They eat the grass. The next year there isn't so much grass The rabbits are hungry, the population decreases. The grass comes back and we cycle around. Ok, so then imagine on another island, a very similar island, we also had some data results of observations and it looked like this. And so here, we see the rabbit population jumping all over doesn't appear to be regular It almost crashes, they almost all die off. Grow fast, almost die off again. And how might we, what might we respond to this Well what would we do with this data. Well one thing that would be natural to say, hey who is messing with the rabbits? We learned from this that rabbits left alone will do their own thing and stabilize and go into this nice cycle. So if we saw some behavior like this it would be logical to think that somebody was messing with the rabbits Something--there must be some external influence crazy weather or random bands of gangs of poachers or you know or who knows what. Bad things to rabbits. People who do bad things to rabbits are coming by. Something is happening to them. But what we have learned in this unit, is that we can get random looking irregular behavior like this without any external influence In fact, the exact same process just changing one little growth parameter by a pretty small amount Can give to rise to both of these situations so I think experimentally One might look at these two and think I have to think about these sort of phenomena in the world very differently That the world is made up of things that are orderly and things that are random. and those are separate. They might get mixed together but they are separate things. And I would need a different type of explanation for this than I would for that. and that is a reasonable thing But chaos says, "No that is actually not true." That you can get disorder from an orderly system deterministic randomness. That randomness and order are not these completely separate things that need to be that we need to think of in a completely different way that in a sense there are two sides to the same coin. So, this suggests that randomness and order that the relationship between them is a subtle than we might have thought. They are not some complete opposites that we need to think completely differently. Here is another way to think about this. So in the prechaos pre-chatoic-dynamic view of things We might be faced with some choices of how we might think about the world or how we think about a different phenomenon. And one choice is the system follows rules We have some rule based system and the presumption and it's a logical one would be that would be if something is following rules it would be orderly. Then the other choice we have is that sytems behavior randomly and when we see randomness we wouldn't explain it by thinking about rules well that's due to chance, stochasticity, or some large amount of external influences. So we would have these two choices and these are very different sorts of ways of thinking about things or explanatory framework. So if we saw something like this We would say "Ah-Ha' it must be following a rule and that's why we see orderliness. Because rules are orderly and rules give-- --orderly rules give rise to orderly behavior. And if we saw this we would say, "Ah-Ha!" The systems behaving randomly its not settling down, it's not cycling So, these rabbits whatever they are doing They are not following a rule It's due to chance Maybe the rabbits behave randomly Some inheriance stochasticy in the affairs of rabbits or there is external influences that are acting irregularly and at random that is making this happen But what chaos and dynamical systems says is that is not necessarily the case. So, we could have a system follow rules. But it doesn't have to be orderly. It could then--we could then have a rule based system that behaves randomly so we might thinks rules imply order we've seen that is not the case logistic equation with r = 4 can produce randomness So rather than this binary choice we have news ways of explaining random or apparently random behavior. We could have a system following a chaotic rule or a chaotic function. So I think this is an important realization for I would think any area of science. So if you see something like this it doesn't automatically mean rulelessness or stochasticity. It could but it doesn't have it. There are other ways we can explain phenomenon like this. So, what does chaos and the butterfly effect have to say about this idea of Laplace and determinism. Well I think that's a really difficult question There are alot of competing ideas and there certainly isn't universal agreement on this. At one level, maybe one of the central premises of Laplace determinism or at least this utonian world view Persists and endures. So, the dynamical system we have been studying in this view the world is still one in sense of cause and effect The world is following rules. Those rules however, don't lead to order or predictability. That sort of twists of chaos and sensitive dependence on initial conditions gives us. In order to do long term prediction of chaotic systems let along come close to Laplace's demon one would need to know the initial condition of whatever it is that we are studying to a degree of percision that is not just impractical but I think is impossible So, I don't know if it spells the end of Laplace's Determinism but it does require us to reconsider those notions alot. Some people argue that the phenomenon on sensitive dependence on initial conditions sort of frees us from the shackles of determinism and gives us a place for free to to exist again. Personally, I'm not so sure. Maybe it frees us from the shackles from determinism sort of But it unclear the butterfly effect, which in a sense is unpredictable shuffling already well shuffled decks of cards and extra time to change your luck so I don't know how the butterfly effect leaves room for free will, but I'm not sure. I'm don't know how to think about free will as I said before. So this brings us to the end of unit 3. This was a long unit. Longer than what I actually intended it to be. But we covered alot of ground but we have learned about the core phenomenon of chaos and the key element of that is sensitive dependence on initial conditions or the butterfly effect. We will explore further and think more about the consequences of this and other phenomenon from dynamical systems in the next several units. See next week in Unit 4.