In these notes I will use mathematical notation similar to the syntax
of Mathematica.
2. Fixed points, January 21.
In order to answer the question about fixed points posed in the last lecture, it helps to look at a cobweb diagram in a small region near a fixed point. The graph of the function f[x] of (1.2) is a parabola, but near the fixed point it can be approximated by a straight line of slope m. In order to understand this situation we can look at cobweb diagrams for linear functions. To do this, download the Cobweb Notebook and replace code defining the function f[x] by
m = .5; l[x_] := .5 + m*(x - .5);
CobWeb[l, 1, .1, 20]
This gives the cobweb diagram for a line with slope .5, shown in Figure 2.1. Figure 2.1 Cobweb diagram for the line y = .5x + .25.
We get a staircase leading to the fixed point. A similar thing will happen if we choose an initial value to the right of the fixed point. If we change the slope to -1.5, CobWeb[l, 1, .45, 5], we get Figure 2.2. Cobweb diagram for the line y = -1.5x + 1.25.
This is a psiral leading away from the fixed point. If the slope is -1, CobWeb[l, 1, .7, 20] gives Figure 2.3. Cobweb diagram for the line y = -x + 1.
In this case we do not move toward or away from the fixed point.
Experiments similar to this lead to the following conclusions about the behavior of near a fixed point:
Definition 2.4. Suppose that we have a function f[x] such as the one defined in (1.2), that x is a a fixed point, ie.e a value satisfying the equation f[x] = x, and the slope of the curve at that point is m = f'[x]. Then we say that this fixed point is
We want to analyze the fixed points of the function f[x] of (1.2) above, with 0 < r < 4. We can use Mathematica to help with the calculations. The code
Clear[r,x]; f[x_] := r*x*(1-x);
FixedPoints = InputForm[Solve[f[x]==x,x]]
gives the answer
Then typing
Simplify[f'[x] /. FixedPoints[[1]]]
gives
This means the derivative of f at the left fixed point (x = 0) is r, and the derivative at the right fixed point (x = (r-1/r)) is 2-r. The behavior at each fixed point depends on this derivative as described above, and si summarized in the following table. Table of fixed point behavior for f[x] = r*x*(1-x).
Fixed point |
x = 0 |
x = (r-1)/r |
0 <= r < 1 |
Attracting |
Repelling |
r = 1 |
Indifferent |
Indifferent |
1 < r < 3 |
Repelling |
Attracting |
r = 3 |
Repelling |
Indifferent |
3 < r <= 4 |
Repelling |
Repelling |
In particular we see that the right fixed point is repelling for r>3. Cobweb experiments indicate that in this case the the population does not converge to the fixed point.
For r slightly above 3, say r = 3.4, the population eventually oscillates between two different values in alternate years. Here is to cobwebdiagram given by CobWeb[f,1.0,.1,20]. Figure 2.7. Cobweb diagram for the parabola y = 3.4*x*(1-x).
How can we understand this? The above analysis of behavior near a fixed point does not help because this behavior is not close to either the left or right fixed point. It helps instead to experiment with the function
which describes the behavior of the population in even numbered years. Here is the corresponding cobweb diagram. Figure 2.9. Cobweb diagram for the curve y = f2[x] with r = 3.4.
Here the population converges to an attracting fixed point corresponding to the larger of the two values shown in the previous diagram.
We can find the fixed points of f2 as before, and they are
{{x -> 0}, {x -> 1 - r^(-1)},
{x -> (r + r^2 - (r^2*(-3 - 2*r + r^2))^(1/2))/(2*r^2)},
{x -> (r + r^2 + (r^2*(-3 - 2*r + r^2))^(1/2))/(2*r^2)}} (2.10)
The first two of these we already knew about, because they are also fixed points of f[x], but the second two are new. Since they involve the sqaure root of r^2-2r-3, they are real only if this quantity is nonnegative, which means r >= 3.
The derivatives of f2[x] at these four fixed points are
Note that the two new fixed points both have the same derivative, 4 + 2*r - r^2. These fixed points will be attracting when this quantity has absolute value less that 1, and this happens when
3 < r < 1 + Sqrt[6] = 3.44949.
This page was last revised on February 2, 1998.