You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.
The logistic models
p(n + 1) = f(p(n))
f(p) = a (1 - bp) p
exhibit some very surprizing behavior. We have seen examples of logistic models that settle down to a limit, that settle down into a 2-cycle or a longer cycle, and we have seen examples that appear to behave chaotically.
In this module we examine the transition from the nicest behavior -- settling down to a limit -- to a 2-cycle. This is one part of the Road to Chaos.
The four movies in this module look at the family of logistic models.
p(n + 1) = f(p(n))
f(p) = a (1 - 0.01 p) p
As the first two movies run a varies from 0 to 4. The first movie shows the first 20 generations in this model starting with
p(1) = 5.
The second movie shows the following
The first 20 generations
The curves y = f(p) and y = f(f(p))
Recall that an equilibrium point is a point at which the curve y = f(p) crosses the diagonal line y = p and that an equilibrium point is attracting if |f'(p)| is less than one at the equilibrium point.
Recall also that a 2-cycle is a pair of points where the curve y = f(f(p)) crosses the line y = p that are not equilibrium points -- that is, where the curve y = f(p) does not cross the line y = p.
We can see an attracting 2-cycle appear for a while in the first movie because after a few generations the population appears to be bouncing back-and-forth between two values. We can see the same attracting 2-cycle appear in the second movie by noticing that the curve y = f(f(p)) does cross the diagonal line y = p twice at points that are not equilibrium points.
We can also see from the second movie why some of these models settle into a 2-cycle rather than a limit. The equilibrium point is not attracting because the slope of the curve y = f(p) (in red) at that point is bigger than one in absolute value. Looking at the curve y = f(f(p)) (in blue) at the two points in the 2-cycle we see that its slope is less than one in absolute value. That is why the 2-cycle is attracting.
We are interested in a phenomenon called period-doubline in which an attracting equilibrium point becomes repelling and at the same time an attracting 2-cycle appears. More generally, an attracting n-cycle becomes repelling at the same time that an attracting 2n-cycle appears.
The second two movies show an example of period-doubling in which an attracting 2-cycle becomes repelling at the same time that an attracting 4-cycle appears. Both movies show the following.
To understand period-doubling graophically look at the three frames below -- for a = 2.7, 3.0, and 3.3.
All three frames have an S-shaped piece of the curve y = f(f(p)) centered around the nonzero equilibrium point q. In the lefthand frame (a = 2.7) this S-shaped piece crosses the diagonal at only one point -- the nonzero equilibrium point q. In the middle frame (a = 3.0) it still crosses the diagonal at only one point, q, but it is tangent to the diagonal line at that point. This is the instant at which the equilibrium point is about to become repelling and the 2-cycle is about to appear. In the righthand frame (a = 3.3) the S-shaped piece crosses the diagonal line at three points -- the equilibrium point q and two new points. These two new points are the two points in the 2-cycle.
We can add to our understanding of this phenomenon by looking at the function
g(p) = f(f(p))
and its derivative and its second derivative.
At the equilibrium point q we see since f(q) = q that
and
Looking back at the frame a = 3.0 or by calculating f'(q) when a = 3.0 we see that in this frame f'(q) = -1.
Thus, g'(q) = 1 and we see algebraically the same thing we see visually in this frame. The curve y = g(p) is tangent to the diagonal line y = p at this point.
Next notice that g''(q) = 0 corresponding to the fact that this point is a point of inflection for the curve y = q(p).
The calculations we have made rely on the Chain Rule and other general rules for differentiation rather than the specific form of the original function f(p). This is a clue to the fact that period-doubling is a very general phenomenon and is not just an isolated phenomenon involving only logistic models. In fact, the whole "road to chaos" turns out to be a very general phenomenon. This kind of behavior rather than being exceptional turns out to be very common.
p(n + 1) = f(p(n))
f(p) = a p / 2^p
where a is a positive constant. Do you observe the same kind of period-doubling for this family of models?