Computer Algebra Recipes

344 CHAPTER 8. NONLINEAR DIAGNOSTIC TOOLS
If d is not equal to zero, then f =ln(d) iscalculated. Ifd = 0, we would obtain
f = ¡1. This latter situation is avoided by setting f = 0 in this case.
> if d0 then f:=ln(d) else f=0 end if;
To perform the sum in equation (8.18), the total is incremented by the value of
f, and the inner loop ended.
> total:=total+f;
> end do:
The counter c is incremented by one,
> c:=c+1;
and a list of points formed with a as the horizontal coordinate and ¸ =
total=numpts as the vertical coordinate. This completes the evaluation of ¸
in equation (8.18) for a given a value.
> pts[c]:=[a,total/numpts]:
> end do:
The sequence of N = 480 points is plotted, the points being joined using a line
style. Figure 8.14 shows the Lyapunov exponent ¸ for a =2:8 toa =4.
> plot([seq(pts[i],i=1..N)],style=line,view=[Sa..Fa,-1.5..1],
tickmarks=[3,3]);
1
0
-1
3 3.5 4
Figure 8.14: Lyapunov exponent (vertical axis) for a =2:8 toa =4.
If one compares the periodic windows where the Lyapunov exponent goes negative,
there is good agreement with the bifurcation diagram, Figure 8.13, for
the logistic map. Because the Lyapunov spikes are sometimes quite narrow, the
agreement can be improved by zooming in on a particular a region by altering
the values of Sa and Fa.