6.41 animation orbit structure (11.4.97)

6.41.1 Aldrovando Azeredo Araujo

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?

6.41.2 Robert Israel (16.4.97)

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, 
# 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).

6.41.3 faia michael a (17.4.97)

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.