Silent introduction This is on the course materials page. Now let's look at a NetLogo version of the logistic map. And what this is going to do is exactly what I was doing in the previous subunit, which is showing the parabola of the logistic map, showing how the trajectory of points moves along it over time. Here I can set R with the slider and I can set the initial value for x. x0, so R is 2, x0 is 0.2000. Here's the logistic map. And I click set up and it draws my parabola. And here you can see I'm plotting x sub t versus x sub (t plus 1). We start out x sub t is point two, x sub t plus one is point three two. And every time I click "go," it advances the system by one time step. Here we plotted x of t is .32, x of t+1 is .4352 and the point moved. So you see .32 on the x axis here and .4352 on the y axis. Go again. And I go again And you can see the whole thing iterating until it has reached a fixed point attractor of .5. And you can see the plot of x_t versus time right here. So these are two different ways of visualizing the trajectory of the map. Now we can also see that, if we change x0... let's take x0 now all the way up to .9 for example. Well, .90285. Close enough. I'll do "set up," Ok. Same parabola, but now x0 starts at a different point. So here's .9029 and the corresponding next time step is .1754 So you can see that's this blue point And we go. And we go again. And you can see that the system goes back to the point .5 and stays there. Same thing if we change x0 all the way down to, let's say, some really small value here .01143. So we "set up" It's very close to zero down here at the bottom. And Go. I click 'go' repeatedly. Again, the system ends up at the value of .5 and stays there. So this .5 here is a fixed point attractor for the system. No matter where the system starts, it always ends up at the value of the attractor. Now, we can vary our value of R to see how the growth rate, R, changes as it moves up. So I'm going to change it here to... it's a little bit hard to change precisely ... to 2.50 I should point out, another way to change this, if you want it to have a precise value, is to right click on it, edit and I can give its exact value here. So I just type that in. Ok, so now what happens? Set up. Well now this time the parabola is higher. And that's because I changed the value of R And what changing the value of R does is it changes the height of this parabola. Ok. So now let's say I start out with x0 = here I'm going to edit it and I'm going to change this value to .2. Ok I'm going to do 'set up' again. Ok now .2 and .4 and we Go ... and we see very quickly the system gets to .6 That was the quiz from the last subunit. And if I change the x0 again, let's make it .9 something, do 'set up' ... Let's see what happens. Again, the system converges on .6 But it does it by a different trajectory. It gets there in a different way. It takes a little longer this time. And again, if I move this all the way down to something very low here... "set up" .. ok now this is close to zero. I do go. Again, it gradually gets to .6 So .6 is the fixed point attractor for this system with a different value of R. So you can see that the value of R is really what determines the ultimate dynamics of the system. And when we talk about the ultimate dynamics, we talk about what kind of attractor, if any, the system settles down into after some number of time steps. So this system has, we say, 'fixed point attractor dynamics' where the fixed point is .6. Ok, let's try this another time. Let's set this R to 2.8 I'll set x again to .2 Ok. Set up. The parabola is higher again. And let's see what happens. Well, this time you can see very ... the little dot bouncing back and forth. It takes a long time for it to settle down And finally it settled down to this point .6429 Remember these are approximations because x_t is a real number and it's possible that there are more decimal places out there that we're cutting off by rounding to a certain number of decimal places. So it's possible that it could still be oscillating more if we looked at more decimal places. But eventually it does settle down into a fixed point. But it takes a longer time. That's one of the effects of raising this R... is the time it takes to settle down to a fixed point. One thing to note about NetLogo is again I can go into one of these output boxes and you can set the number of decimal places you want to see. And the maximum it allows you is seventeen.