Computer Algebra Recipes

340 CHAPTER 8. NONLINEAR DIAGNOSTIC TOOLS
and the outer do loop ended. The N = 200 plots are superimposed with the
display command, yielding the bifurcation diagram shown in Figure 8.13.
x
> display([seq(Gr[m],m=1..N)],view=[Sa..Fa,0..1],
1
0
labels=["a","x"]);
3 4
a
Figure 8.13: Bifurcation diagram for the logistic map for a =2:9 to4.
Starting at a =2:9, the reader can observe period 1 occurring up to a =3:0.
Then the steady-state response undergoes a so-called pitchfork bifurcation to
period 2, followed by clearly seen bifurcations to period four, period eight, and
a barely observable period-sixteen solution. At higher a values, the response is
generally chaotic, but narrow periodic windows also occur. The reader should
be able to see, for example, a period-3 solution for a ¼ 3:83. In this case, the
bifurcation to period 3 is an example of a tangent bifurcation.
Since the periodic windows are often very narrow in terms of the range of a,
one should really increase the number N, but this leads to a longer computing
time. A better approach is to leave N unchanged and zoom in on a particular
range of a by changing the values of Sa and Fa.
Bifurcation diagrams are very useful diagnostic tools for studying the behavior
of nonlinear maps as well as forced oscillator ODEs as one or more control
parameters are varied.
A complementary graphical approach is to calculate the Lyapunov exponent
as a function of a as illustrated in the next recipe.