So let's summarize Unit 4, which was the first of two units on bifurcations.
I began by looking at the logistic differential equation.
So looks very similar on the right-hand side to the logistic equation that you have iterated, but this is a differential equation.
As before is a very simple model of a population growth where there is some limit to growth.
The quantity r is the growth parameter, k is known as the carrying capacity and the reason for that is that we saw that there was a stable equilibrium at P = K.
So that's the equilibrium value to it the population will be attracted.
So there is a stable equilibrium or attractor at P = K and an unstable equilibrium or repeller at P = 0.
So then I compared and contrasted differential equations, that's described here and iterated functions.
So for iterated functions, that's what we worked on Unit 3 mostly.
Time moves in jumps and the population moves in jumps.
The population might jump from 0.6 to 0.8 without going through any of the in-between values.
For the differential equations of this chapter or unit, time is continuous and P, the population, is continuous.
So if I know that one year of the population was eighty and another year was a hundred, so it had to have gone through all the intermediate values.
All values between eighty and a hundred.
So this fact of continuity together with determinism means that cycles and hence chaos are not possible for one dimensional differential equations of this sort.
And again this is due to determinism, because for a given P, a given population value, the population can have only one derivative, only one rate of change.
So it can't cycle around, go up and down, for the same P value. It has to either go up and down.
So P can only increase to a fixed point, decrease to a fixed point or tend towards plus or minus infinity.
So the range of behaviors we can see here is a little dull in the sense that we don't see chaotic behavior or even cycles.
For the iterated functions that we have studied in unit 1 and 3, cycles and chaos are possible.
And I should mention that both the differential equation and the iterated function are sometimes called logistic equation.
But iterated version of it is often called the logistic map an the picture here is that the function maps the unit of interval to the other... to the unit of interval... maps the unit of interval to itself.
So it a sort of technical sense of the word mapping.
But anyway, if you hear the term logistic map, it's almost surely an iterative function. If you just hear the term logistic equation, it could be this or that.
It usually will be clear from the context, which one is which.
So then we look at a modification of the logistic equation. I added a harvest term h, so the population is growing doing its own thing, but each year we subtract h fish from the population, so h is for harvest.
And we asked what happen for different values of h. So here is one of our first examples where h = 40 and this parabola shifts down and so the fixed point that used to be here moved in and here is the phase line for that situation.
We had a stable fixed point at 84 and an unstable fixed point around 16.
So we asked what happen for different values of h.
And we took a couple of snapshots. We did h = 40, h = 60, h = 20. h = 80. I forget the exact values, but we have experimented with half of dozen or so different h values and for each experiment that we did, drawn a phase line and then we combined all those phase lines and others in this construction called the bifurcation diagram.
So here's the bifurcation diagram for a possible... We had the general shape that we would expect for the logistic equation.
So we took the phase lines and for different parameter values and then sort of glued them together.
And so in a bifurcation diagram the parameter is changing this way and in the vertical axis, where each slice give us the phase line.
So these are a little abstract and can take a little of practice to get used to interpret them, but the key is to remember that a bifurcation diagram is just a lot of phase lines glued of stacked together.
So if you want to know, say, what's going on for a little bit above a hundred, this case, I would try to just focus in right on that vertical slice and I would see that there is a stable fixed point here and an unstable fixed point here.
So bifurcation diagrams show how fixed points of a dynamical system chance as the parameter varies and we could have changes where a fixed point changes its stability or we could have something where a fixed point disappears all together and that's the case here.
So bifurcation diagrams are very common graphical tools used in the analysis of dynamical systems. When one presents an analysis of a model, often one summarizes that with a bifurcation diagram. So they are a very useful geometric tool.
And we will revisit these in the next unit when we will look at bifurcation diagrams for the iterated function systems. The logistic equation iterated.
So then we turned from bifurcation diagrams to bifurcations themselves. So a bifurcation is a subtle qualitative change in the behavior of a dynamical system as the parameter is varied.
And what I mean by a qualitative change is a change in the number of fixed points and/or change in its stability.
So here we have a bifurcation in this diagram like around two hundred and so if we were above 200 we had no fixed points and if we were below 200 we would have two fixed points, one stable, one unstable.
So the moral of the story is that sometimes properties of continuous models are discontinuous.
So let me explain what I mean by that, just to review.
If I map here and I increase h, a small change in h leads to a small change in the value of this equilibrium, this attractor. If I'm here and I do a little change in h, a small change in h gives rise to a small change in the equilibrium value. If I'm here however and I do a small change in h, I increase h a little bit, so I fall off, this attractor disappears and the population would suddenly crash.
There is not any stable populations between these two values, so there is a gap and if you move these way there would be a jump.
So most of the time a small change in your model leads to a small change in the behavior, but sometimes it can lead to a sudden jump.
So that's an important realization, the most important realization in this chapter, that continuous models well-behaved differential equations can sometimes have these sort of jumps in them.
So lastly I'm mentioning briefly and I'll do so again, there is a nice classification of bifurcations into several different types. This is a little beyond the scope of what I want to do in this course, but if you want to dig deeper, the reference I recommend the most would be chapter 3 of Strogatz's book, the entire book is great and this chapter is really nice. It's just one or two notches more mathematical than what you are doing in this course.
Scholarpedia, the URL is here, has a pretty good set of pages on bifurcation.
Wikipedia set of pages are pretty good as well. I think Scholarpedia's might be a little bit more readable.\
And this is a standard topic, so most texts on dynamical systems or differential equations, modern differential equations book, would have some discussion on bifurcations in them.
So there are lots of other places you can go if you want to dig deeper into this phenomena of bifurcation and bifurcation diagrams.
So this brings us to the end of Unit 4. In the next unit we will continue to learn about bifurcation diagrams and we will be in for some fun surprises when we look at the bifurcation diagram for the iterated logistic equation.
See you next week.