Computer Algebra Recipes

8.1. THE POINCAR ESECTION 321
for the ith solution, the evaluations being carried out in the following do loop.
> for i from 1 to N1 do xx[i]:=X(i); yy[i]:=Y(i); end do:
An operator Gr is created to graph a phase-plane point, represented by a size-16
blue cross, at time t = nT0 for the ith amplitude.
> Gr:=(i,n)->pointplot(f[yy[i](n*T[0]),xx[i](n*T[0])]g
color=blue,symbol=CROSS,symbolsize=16):
Using this graphing operator in the following loop generates the Poincare section
for each of the N1 = 4 amplitude values. To eliminate the transient part of the
solution, a certain number of initial points must be removed in each plot. This
number will vary for each numerical run and must usually be determined by
trial and error. Here, the ¯rst 24 points have been removed. Since N2 =250,
each plot still contains 225 points. A suitable viewing box is selected, which is
the same for all four plots, and labels added and the minimum number of tick
marks controlled.
> for i from 1 to N1 do
> display([seq(Gr(i,n),n=25..N2)],axes=boxed,view=
[-0.4..0.8,-1.4..1.4],labels=["y","x"],tickmarks=[3,3]);
> end do;
1
x
–1
0 y 0.5
1
x
–1
0 y 0.5
Figure 8.1: Period-1 (left ¯gure) and period-2 (right) Poincare sections.
The plot on the left of Figure 8.1 corresponds to F1 =0:325, the one on the right
to F2 =0:35. Despite the fact that 225 points were plotted, we see only a single
cross in the F1 =0:325 graph, indicating that the system has settled down to
a period-1 solution. For F2 =0:35, the ODE system oscillates back and forth
between the two crosses, the Poincare section being characteristic of a period-2
solution. The observed periodicities agree with the results in Section 1.2.1.
For F3 =0:356, Figure 8.2 indicates a period-4 solution, again agreeing
with our earlier conclusion about the periodicity for this forcing amplitude.
So it is clear that the Poincare section approach is a useful graphical tool for
interpreting the periodicity of driven oscillator systems.