#### 6.41 animation orbit structure (11.4.97)

I am very new to Maple and maybe my problem is easily solved. Anyway, I am working with
diﬀerential equations with a geometrical approach. So I am more interested in analysing the
orbit structure of diﬀerential equations.

Is there a plot procedure that gives me the animated orbit of some initial condition in the
space, like following some orbit of Lorentz’s equation into the attractor? For example like the
"comet" function in MatLab?

I don’t know about "comet", but here’s one thing you can do. This requires Release 4 (but
there are ways to get similar eﬀects in Release 3).

> with(plots): with(linalg):
> lorenz:= D(x)(t) = sigma * (y - x), D(y)(t) = rho * x - y - x*z,
D(z)(t) = - beta*z + x*y;
> sigma:= 10; rho:= 28; beta:= 8/3;
> ics:= x(0) = 1, y(0) = 1, z(0) = 1 ;
> vals:= dsolve({lorenz, ics}, [x(t), y(t), z(t)], numeric,
value=array([seq(.01*n, n=0 .. 2999)]), method=classical[rk4]):
# this calculates 3000 points on the trajectory: may take a while
> vmatrix:= delcols(vals[2,1],1..1):
> vlist:= convert(vmatrix, listlist):
# vlist is a list of 3000 points
> traj:= spacecurve(vlist):
# this is the trajectory, plotted as a curve
> for j from 1 to 20 do
frame[j]:= pointplot3d([ seq(vlist[20*k+j], k = 0 .. 149)]) od:
# this produces 20 frames, each consisting of a plot of 150 points on the
# trajectory
> display({ traj,
display([seq(frame[j],j=1..20)], insequence=true,
scaling=constrained)},orientation=[80,80]);
# this produces the animation. Click on the plot to get "VCR controls" on
# the Context Bar. Click the Loop button (the one that looks like an oval with
# an arrowhead), and Play (the triangle pointing right).

There is a section of Nicolaides-Walkington that does this. Incidentally, I believe Lorenz is
spelled without a t. N-W make the same error.