Computer Algebra Recipes

352 CHAPTER 8. NONLINEAR DIAGNOSTIC TOOLS
¢:= 1
40
The system of ODEs is solved numerically, the solution being given as a list
procedure, so that we can create a time series.
> sol:=dsolve(fsys,icg,fx(t),y(t),z(t)g,numeric,maxfun=0,
output=listprocedure):
The following line uses the numerical solution to evaluate x(t) atanarbitrary
time, which must be speci¯ed.
> S:=eval(x(t),sol):
Then entering S(n*Delta) will yield x(t) att = n ¢. Using this result, the x
values are obtained at times t = n ¢forn =0toN.
> for n from 0 to N do X[n]:=S(n*Delta); end do:
The time series is created and plotted, the default being to join the points with
straight lines. The resulting picture is shown in Figure 8.20.
> plot([seq([n*Delta,X[n]],n=0..N)],labels=["t","x"]);
x
15
10
5
0
–5
–10
–15
10 20 30 40 50
t
Figure 8.20: x time series for the Lorenz system.
To reconstruct the 3-dimensional Lorenz butter°y attractor, it is necessary to
form triplets of numbers from the time series. By trial and error, we use the
triplet combination involving n, n +3,andn + 6 to create the plotting points.
> points:=[seq([X[n],X[n+3],X[n+6]],n=0..(N-6))]:
Using the spacecurve command with shading=z to color the trajectory, the
butter°y attractor is revealed.
> spacecurve(points,style=line,shading=z,axes=framed,
orientation=[-30,60],tickmarks=[3,3,3],
labels=["X(n)","X(n+3)","X(n+6)"]);
A black-and-white version is shown in Figure 8.21. If one compares the picture,
which can be rotated on the computer screen, with the butter°y picture
obtained in Chapter 2, the reconstruction of the butter°y is quite good.