In this video, I'll compare and contrast differential equations and iterated functions. These are the two main types of dynamical systems that we'll study in this course, and, although they're very similar, they do have some different mathematical properties, and comparing the logistic equation, as an iterated function and differential equation can help make this clear, and highlight some important distinctions. So here on the left, is the logistic differential equation. A differential equation (the form we'll be studying) describes a function P in terms of its rate of change. So this says, we know the rate of change of P if we know what P is. The population growth depends on the population value in these two parameters. For an iterated function, it also describes a population growth, but here, f(P) is the population next year, given the population P this year. So we get a series of population values by iterating this function. So I began when I derived the logistic equation, (I used this form), but it's often simplified to this, the A kind of gets absorbed inside x. So this is what we worked with, but the starting point for these two equations is the same on the right-hand side. What's different is, we interpret things differently on the left-hand side. So the right-hand side here is interpreted as the growth rate. The right-hand side here is interpreted as the population next year. So solutions to these iterated functions and differential equations have a different character. For differential equations, the solution is population as a function of time, and that would look, as we've seen, maybe something like this. For an iterated function, we end up with a time series plot, and that might look something like this. So notice the difference between these two solutions. In both cases, the blue curve is the solution to the dynamical system. The dynamical system is just a rule that tells this blue thing what to do. But for the differential equation, the blue curve changes continuously. It's defined at all times, and it smoothly increases, say, from here to here, and it has to pass through all intermediate values. For the iterated function, the time moves in jumps. It has an initial value at time 0, then time 1, then time 2, and the value of the population also moves in jumps. It goes from this value to this value, and even though we connect those dots, it doesn't slide through all values in between. It jumps from here at time 0 to here at time 1 without going through the intermediate values. In this one, the differential equation, time and the population are continuous. Time and population are continuous. But for the logistic equation and all iterated functions, the time and the population or whatever we're measuring moves in jumps. So, again, for the logistic equation and the iterated function, time and population moves in jumps. And this difference here, together with the fact that these equations are deterministic, gives rise to very different ranges of possible behaviors. So we've seen for the iterated function in Unit 3 that it's capable of producing cycles and chaos. So cycles and chaos are possible. Of course not all iterated functions will show a cycle or will show chaos and remember chaos is an aperiodic bounded orbit that also has sensitive dependence on initial conditions. For a differential equation, however, cycles and chaos are not possible. So let's think about why this is so. So suppose a cycle was possible. If that was the case, I would have a solution curve that looked something like that. It goes up and down. We can eliminate this possibility by appealing to the determinism of this equation. This equation says that the derivative, the growth rate, the rate of change of the population, depends only on the population. (And r and K, but we're imagining those are fixed.) So let's think about this blue curve here that oscillates up and down. I'm going to draw, just arbitrarily, a dashed line through here. And notice what happens. Here, I have a particular p value, the p value is at this dashed line, and the population is increasing so the derivative is positive. The derivative is positive for this p value. Over here, when the population is going back down, the population is decreasing, so the derivative is negative. So that means at these two points, here and here, they're different derivatives. So at the first purple arrow the function is increasing : positive derivative. At the second arrow the function is decreasing: negative derivative. But the problem is that they have the same p value as on the y axis here. And the p value is the same. If this was true, this would say different derivatives at the same p value. But that's impossible because the differential equation says the derivative is a function of only the p value. Another way of saying that is a given p value only has one derivative associated with it. If you know the population p then that determines the derivative. Here, if you know the population p that does not determine the derivative, because you have different derivatives at the same p value. So the conclusion, then, is that cycles are not possible, and chaos isn't possible as well. Any behavior that goes up and down (it doesn't have to be a regular cycle) we can eliminate by this argument. As we said in Unit 2 the range of behaviors for one dimensional differential equations are kind of boring. The function can increase to a fixed point decrease to a fixed point, decrease to infinity, increase to infinity, and that's all it can do. Iterated functions have a much richer array of behavior, and that's because determinism doesn't constrain them in the same way, so that it doesn't forbid cycles. So cycles are possible in iterated functions and chaos, aperiodic behavior, is possible as well. In the next sub-unit we'll leave iterated functions behind for a little bit and we'll look again at the logistic differential equation and I'll add a term to it and we'll start investigating bifurcations.