Fixed Point Iteration: Logistic Function
Mth 351 June 30 2002 Maple 6
Bent E. Petersen
Filename: 351u2002_logistic.mws
Assignment 1. Due July 8, 2002. See the two problems at the end of this worksheet.
> restart;
We will investigate t he fixed p oint it erat ion for t he lo gistic function
> f:=(x,c)->c*x*(1-x);
:= f → (),xc cx() − 1x
Here c is a parameter which satisfies 0 <= c <= 4. For these values of c we have 0 <= f(x,c) <= 1
for each x satisfying 0 <= x <= 1, that is, as a function of x, f(x,c) maps the interval [0,1] into
itself. T hus by the I ntermediate Value Theor em ( IVT) f(x,c) is guaranteed to have a fixed point in
[0,1]. This fact is not particularly useful here though since the origin is an obvious fixed point, and the
IVT doesn’t give us any additional fixed points.
It is not difficult to see that f(x,c) is a contraction map for 0 <= c < 1. In this case we have a unique
fixed point and the fixed po int iterations will converge very rapidly to the unique fixed point , no matter
what init ial "guess" we st ar t with. For other values o f c it is not clear how t he fixed point iterates will
behave.
Our goal here is to plot some iterates for various values of c and attempt to describe what we observe.
Even in the case of convergence to a fixed point, the first few iterates may depend heavily on the initial
point and so may be all over the place. Therefore we discard the first M0 iterates and plot only
subsequent iterates.
Her e are the values of t he parameter c that we will wor k with. We omit 0 because it is not interesting.:
> N0:=30; # N0 values of c
:= N0 30
> Cvals:=[seq(4*k/N0, k=1..N0)];
Cvals 2
15
4
15
2
5
8
15
2
3
4
5
14
15
16
15
6
5
4
3
22
15
8
5
26
15
28
15 232
15
34
15
12
5
38
15
8
3
14
5
44
15
46
15
16
5
10
3
52
15
,,,,,,,,,,,,,,,,,,,,,,,,,,
:=
18
5
56
15
58
15 4,,,
