In this subunit we will explore the bifurcation diagram for the logistical map further and deeper. To do so, we will use the web program to zoom in on different regions of the bifurcation diagram. Before doing that, I want to remind you once again a bifurcation diagram is formed. So, a bifurcation diagram is a collection of final state diagrams. So we need to make alot of final state diagrams. For each of those final state diagrams in the collection, we need to choose r Iteratre for 100 time steps 100 iterations. Then, we'll plot the next 200 iterations. Then you will repeat for a bunch of different r-values. And then, we will want to glue together. Line up all those final state diagrams and you will get a bifurcation diagram. Which, would give you a glimpse the different behaviors possible of r values. As, we zoom in to the bifurcation diagram, to look more closely at some of its structures and patterns. We are going to change these numbers. Sometimes we will need to ask the computer to iterate for more than 100 times. Maybe a thousand or maybe more. And other times, for a different reason, we are going to need to plot than 200 points. So, this number 100 and 200, those are the defaults I've been using so far. But, we need to change these, when we do our explorations. So, let's get started and look at the bifurcation diagram and start zooming in and seeing what we see. Here is the bifucation diagram program and I resized both the browser window and the image size. So it fits well in the screen, so let's start zooming in. The interesting action happens here on the right. So, I'll zoom in. And, there is a close up view. And, I don't know Let's say I want to see. What's going on in this region, here in the middle Opps. So let's zoom in on that region And I'll zoom in once more. So again I'm seeing this motiff, this pattern We see occur again and again. But as I zoomed in so much, in a sense I've lost resolution. There aren't that many points here. And the reaon for that is the x range is now very small. I'm going 0.78 to 0.488 aproximately. So I"m seeing just a very tiny slice. A very tiny proportion of a long final state diagram. So, for each r value, I've plotted 200 points. But most of the points fall off the screen, and they are above an below this little slice. So I need to do something so I can see more points. I want to see what the pattern looks like in here. And so, I'm going to increase the number here. I'm going to skip 100, but rather than plotting the next 200 I'll plot the next 2000. So let me update that. And pretty quickly I can see the image is alot darker. That means there is alot more points, that means I can see the structure. I can see what's going on there. So, this affect the image becoming grainy or losing resolution. It is if, if you are zooming in on an old analogue photograph and you literally run out of resolution. You start to see individual grains or pixels. So the strategy here, once you start to zoom in and lose dots is to ask the program to plot more dots. It's important though when doing the zooms to not ask the program to plot too many dots too soon. The reason for that is, is that the slowest part of the program is the plotting. and what I mean is the rendering of each of these pixels on the screen. That's the slowest step. The computer when it does its iterations is a very fast process. So, here I'm asking the program to plot 2000 for each R value. But, it doesn't actually plot 2000 when I zoom in. Because, most of the iterates for a particular r value are going to fall outside this narrow interval. And so, asking it to plot these extra points. Doesn't slow the program down too much. So let me illustrate that as follows. Let's say I got really excited and I wanated it to do 20,000 points. So calculate 20,000 points and plot them. This will take a little bit of time, but still not much. Let's give it a try. One, two ,three. Sot hat took about three seconds. And by the way this is-- --if I wanted to look at this, understand the image, this is too many points. It's a little bit like an overexposed photograph or something. But in any event asking it to do all this iteration it does about 100 r values, then iterating for another 20,000 intervals took about three seconds. And that's because I wans't asking it to plot that many. Let me go back to-- --let's say I go back to this image. If I asked this to do 20,000 it's going to take a long time. Let's say how long-- --it might take too long. Let's give it a try. 1,2,3,4. Alright so that took about four seconds longer than it did before. Let me go bak a little bit more. Let's try this. I just want a shot. 1,2,3,4,5... ...26,27,28. That took almost thirty seconds. So the moral of the story is, when you zoom in you start to lose resolution because there aren't enough points plotted. Because there aren't enough points plotted. So, you want to ask the program to plot more points for you. However, don't ask the program to plot more points for you until you need it. Because plotting takes a long time. And the program will become very slow to work with. Ok, so as we zoom in, we need to ask the computer to plot more points. And we don't want to plot too many points. We don't want to ask it to plot too many points too soon. There is another issue or subtlety that arises when zooming on the bifurcation diagram that's the different from losing resolutions, not having enough points issue. That I talked about previously. So let me illustrate this, and I'll zoom in here. And see what's going on. So this is where the bifurcation is from period 2 to period 4 occurs. And maybe I want to look right in at that point. Do that. And we start to see some weird structures here. Some, I don't this almost looks like feathering on my screen. I'm not sure what it will look like on the video. And so, going back we expect to show this analytically. We are going from period 1 to period 4. We see periodic behavior as a single narrow line. A single dot on the final state diagram. But here right here where the bifurcation occurs The single point that we would expect to see. For period 2 or period 4 these single narrow lines. They fuzz out, the blurr out here. We can see why that's the case. Let's go look at the time series plot for this r-value right here. 3.44568. So here is the program that makes time series plots for the logistical equations. And I want to know what's going on for the r value 3.55568. That's where we are starting to see this surprising fuzz or smearing in the bifurcation diagram. So let's make the time series plot and there it is, and maybe it's a little hard to see. But, maybe it's period two. So up-down, but it's actually period 4. So there is a little bit of difference between these two lower points. And, if I were to, if I were to plot this on a bifurcation diagram. --Sorry, a final state diagram. I would have actually four points. Two up here. Two down there. And they will be very close together. However, in a final state diagram. which is what bifurcation diagrams are made of We are interested in the final state the long term behavior So, to check let's see what's going on for larger numbers of iterates. So again I'm going to see there is a period of like period 2, but not quite, There is a period of wiggle in here. But, maybe eventually it is going to become period 2. Here am I plotting 1000 points. And this point it's really hard to see figure. Let's go down here, and look at just this last couple numbers. If I wait for 1000 time steps. So I have to iterate for a long time. Then it actually becomes period 2. I become .44085, .84937, .44085. It's gone back again in a period two cycle. So, when r is 3.44568 in the long run it is really period 2 not almost period 2, but period four. But in the long run it's a pretty long time. If I just wait for the 100 iterates. That we've been doing before. It looks like period 4, when it actually is period 2. Another way of saying this is that the transitory or temporary behavior for this is very long lived. It takes a long time to reach the final state. Another way, yet another way of saying this that there is a period 2 antractor but it's not that strong of an attractor, It doesn't pull in orbits very quickly. In this case it takes 1000 iterations in order for the, for the orbit to reach the attractor. So, this means that we will have to plot, have the program when we do the bifurcation diagram have the program skip plotting more iterates in order to see the long term behavior. So, let's go back to the bifurcation diagram program. And see what we can do. So here we are at the bifurcation diagram program again. And, I'm going to change this number here. So, skip ploting first 100 iterations instead I'm going to ask it to skip plotting 1000 iterations. So what the program will do now for each r-value the program will computer 1000 iterates, then it will plot the next 200 iterates. So I'm changing the to 1000, let's update the plot and all of a sudden we have seen the smugginess has gone away. So here we are seeing this almost period two but actually period four behavior as it wiggles around back and forth not quite hitting the period two attractor but if we wait longer Woops, sorry. If we wait longer, then the period two attractor gets reached and this thing clears itself up I could however, zoom in here again and now again, I see this smugginess or smearing And, again this indicates this again is not reaching an attractor and I'm going to change this, so I skip plotting 10,000. So now we are asking the program to calculate 10,000 iterates for each r value ignore those and then plot the next 200. If we do that this is what it looks like. Again, the smugginess goes away if we late longer and we see this the next period two to period four. So again as we zoom in on the bifurcation diagram to look deeper and deeper to prob it's inner structure. We are going to need to sometimes ask it to skip more iterations and sometimes ask it to plot more orbits. So we will do that in the next video.