Computer Algebra Recipes

8.3. THE BIFURCATION DIAGRAM 339
After a short transient interval, the logistic system oscillates back and forth
between the two branches shown in the ¯gure. This is characteristic of a period-
2 solution, there being only one branch in steady state for a period-1 solution.
As a is increased, further bifurcations occur.
To create a bifurcation diagram for the logistic map, a particular initial
value of x0 is chosen, and the map iterated for a given a. A certain number of
the initial points are thrown away to remove the transient part of the solution,
and the subsequent steady-state points plotted. Then, one increments a by a
small amount and repeats the process, and so on. The whole process is easily
automated, as is now illustrated.
> restart: with(plots):
As you may con¯rm by running the ¯rst part of the recipe, a period-1 solution
prevails until a =3:0, at which point period two begins. In the recipe, the range
of a is taken from the starting value Sa =2:9 uptothe¯nalavalue Fa =4. If
this a range is divided into, say, N = 200 equal intervals,
> Sa:=2.9: Fa:=4: N:=200: stepsize:=(Fa-Sa)/N; x:=0.2;
stepsize := 0:005500000000 x := 0:2
the stepsize is ¢a =0:0055. The initial value of x has been taken to be x =0:2.
A total of 500 iterations will be considered and the ¯rst 100 points ignored
in order to remove any transients. This leaves 400 points to be plotted for each
a value. The quantity c is a counter to keep track of the plots, a graph being
produced for each a value. The counter is initially set equal to zero.
> totalpts:=500: ignorepts:=100: pts:=totalpts-ignorepts; c:=0:
pts := 400
In the following double do loop, the ¯rst, or outer, loop increments a from the
starting value Sa to the ¯nal a value Fa in incremental steps given by stepsize.
> for a from Sa to Fa by stepsize do
The second, or inner, do loop iterates the logistic equation from 1 to totalpts.
> for n from 1 to totalpts do
> x:=a*x*(1-x);
The points for a given a value are formed into a list,
> pts[n]:=[a,x];
> end do;
and the inner do loop completed. Then, the counter is advanced by one,
> c:=c+1;
and a list of lists is made using the ¯nal 400 (presumably) steady-state points.
> points:=[seq(pts[k],k=ignorepts..totalpts)]:
Aplotismadeforeachavalue using a point style,
> Gr[c]:=pointplot(points,symbol=point,color=black):
> end do: