Computer Algebra Recipes

1.2. THREE-DIMENSIONAL AUTONOMOUS SYSTEMS 45
trajectory that results if the initial condition is near the other ¯xed point is left
as a problem.
> ic:=[x(0)=0.1,y(0)=0.1,z(0)=0.1]:
To enter the relevant ODEs, the dependent variables x, y, andz must be made
time-dependent. This is done in the following command line.
> vars:=fx=x(t),y=y(t),z=z(t)g:
Substituting the variables into P , Q, andR, and equating to dx=dt, dy=dt, and
dz=dt, yields the RÄossler system of ODEs.
> sys:=diff(x(t),t)=subs(vars,P),diff(y(t),t)=subs(vars,Q),
diff(z(t),t)=subs(vars,R);
sys := d
d
x (t) =¡y(t) ¡ z(t); y(t) =x(t)+0:2y(t); dt dt
d
z (t) =0:2+z (t)(x (t) ¡ 5:7)
dt
Choosing the option scene=[x,y,z] in the DEplot3d command, we plot the
trajectory in x-y-z space over the time interval t = 0 to 150, subject to the
given initial condition. The step size is taken to be 0.01 in order to obtain a
smooth curve. The trajectory is colored with the zhue shading option, and a
particular orientation of the viewing box is chosen.
> DEplot3d([sys],[x(t),y(t),z(t)],t=0..150,[ic],scene=[x,y,z],
stepsize=0.01,shading=zhue,orientation=[-120,60],
tickmarks=[3,3,3],thickness=1);
20
z
10
5
y
0
–5
–10
Figure 1.18: RÄossler's strange attractor.
0
x
10