So we now have a collection of final state diagrams. They’re the examples that I did in the last video 3.2 , 2.9 and then an aperiodic value at 3.8 and then for the quiz that perhaps you just took you did 3.4 that’s another period 2 value 3.739 this is period 5 1,2,3,4,5 1,2,3,4,5 and then another aperiodic one 3.9 for here the dots extend from about .1 to almost 1 exactly, may be .98 so it might look something like this so we got a collection of final state diagrams and as we did with differential equation we’ll form a bifurcation diagram by gluing together a collection of final state diagrams so let me take each of these and cut it out so I can move it around so how do you that and we’ll see what it looks like so now I’ve got a collection of final state diagrams and let’s put them in order and as before put them on the side as they do that so there is 2.9 that was period 1 here’s 2.3 that’s a stable cycle of period 2 here is 3.4 3.739 3.8 and 3.9 so this is the beginning of a bifurcation diagram for the logistic equation remember the goal of the bifurcation diagram is to see how the dynamical systems behavior changes as a parameter in this case r is changed so it gives us a global view of the range of behaviors that dynamical systems can exhibit in this case with just these six final state diagrams it’s not really clear quite yet what the overall pattern might be so in order to sort of see a pattern to connect the dots so to speak we would need to try this out from many many more r values and make many more phase lines and stack them all in here densely so that we can see what happens from one r value to the next so as you’ve probably guessed I’ll use a computer to do that work for us and I’ll show you the program and how it works in a little bit but first let’s focus on what the results are so let’s see put these to the side just for a moment here’s the bifurcation diagram for the logistic equation the lower limit here is r equals 0 and here’s r equals 4 and then this goes from 0 to 1 so when r is between 0 and 1 between my fingers here, the, can’t quite see it here but the only fixed point is the attracting fixed point is 0 if the growth rate is less than 1 the rabbits die out between 1 and 3 there is an attracting fixed point and in fact we saw that let’s see we did one for 2.9 here it is let’s see if this is going to work, close, ok so 2.9 that’s about there and this dot that I drew in the first example is part of this line here then as r is increased as the growth rate gets larger and larger the period 1 behavior splits into period 2 and we’ve seen that here’s r equals 3.2 r equals 3.2 so the two dots from the final state diagram show up as part of this line here so in this region where if I go up from a single point I see two lines two dark regions that would indicate that its period 2 here is another r value a little bit larger 3.4 and still period 2, there are only 2 dots but the periods are little bit further apart see if I can get both of these on at the same time so for these two different values, it’s the same qualitative behavior attracting cycle of period 2 but the exact locations are a little bit different all right it’s little hard to see what’s going on in here so we’ll zoom in here in just a moment but first just a little bit of terminology which should be familiar from what we did with differential equations I’d say the system undergoes a bifurcation here r equals 3 remember a bifurcation is a sudden qualitative change in the behavior of a dynamical system as a parameter is varied continuously so the qualitative change here is that the fixed point here splits into two so we go from an attractor of period 1 to an attractor of period 2 so that’s a bifurcation and it’s called a period doubling bifurcation because the period doubles here we see we have a bifurcation from period 2 to period 4 so that’s another period doubling bifurcation ok, let’s zoom in on the bifurcation diagram, let’s look at just to this portion, let’s look at what’s going on from 3 to 4 since this is where a lot of the interesting action is so here I’ve zoomed in and this is a bifurcation diagram from 3 to 4 so we see in this region from 3 to about a little more than 3.4 the behavior is period 2, here are the two phase lines we final state diagrams we drew previously there’s 3.2 and there’s 3.4 and they line up pretty well let’s see if I can get a few more on here here’s 3.739 and that corresponds to this funny region here this light region we’ll look at that more closely in a bit but period 5 1,2,3,4,5 and then we had 2 aperiodic values at 3.8 and around there that looks pretty good and then 3.9 which is right around there so the bifurcation diagram for the logistic map looks quite different then the ones we saw for differential equations which isn’t surprising the logistic the logistic map and things like it exhibit chaos aperiodic behavior so we’ll expect it to be more richer bifurcation diagram and have more features to look at but remember the thing about bifurcation diagrams to interpret them remember that they began their life as a series of in this case final state diagrams so for example if I wanted to know what’s going on right around 3.7 I would just try to blot out everything except for 3.7 and then view it as a single final state diagram sort of imagine doing that with this, thing that I’ve made so this is I’ve moved this so that the split shows right around 3.7 and so we would say that ahaa this looks like an aperiodic region lots and lots of dots so it must be aperiodic going from between this value and this value if I want to know what’s going on at 3.2 I could move this until I’m seeing 3.2 and then I would see just these 2 dots here or small line segments and that would mean that this is periodic with period 2 you can imagine another way to view this as r increases you see period 2 behavior and the two values are getting further apart they’re moving this way as I let r get larger and then a little passed 3.4 that’s where is it there it is, there’s a bifurcation, so now it’s period 4 1,2,3,4 like small change I go then a small change in r that’s moving this leads to a qualitative change in the behavior of the dynamical system in this case we go from 2, a cycle of period 2 to a cycle of period 4 and then as I increase r further still there’s a region of period 8 1,2,3,4,5,6,7,8 each period splits into 2 so 4 goes to 8, 8 goes to 16 and so on then we have regions of chaos here this is aperiodic but with a gap in the middle this is it’s very narrow but this is the period 5 value we saw before more aperiodic regions, here’s a period 3 gap 1,2,3 I think we investigated that maybe back in Unit-2 and then finally up at r equals 4 we have orbits that go from 0 to 1 so it would fill this entire interval ok, so this is the bifurcation diagram for the logistic equation we’ll spend lots more time exploring this but first I would recommend doing the quiz it should be quick and it will just kind of check your understanding of this lecture and then we will look at an online program that will let you do much much more exploring with the bifurcation diagram for the logistic equation