Computer Algebra Recipes

1.2. THREE-DIMENSIONAL AUTONOMOUS SYSTEMS 35
the resulting picture being shown in Figure 1.12. The potential is commonly
referred to as the double-well potential. There are two minima, at x = ¡1 and
x = +1, at which points V = ¡ 1
4 , separated by a maximum at x =0,where
V = 0. In the absence of any driving force (set F =0)ordamping(° =0),the
two minima correspond to vortex points, and the maximum is a saddle point.
In this case, the spring system will oscillate in one of the two potential wells
provided that the total energy is less than zero. For a total energy greater than
zero, the oscillations will be back and forth between the two potential wells.
These possible motions can be con¯rmed by making a phase-plane portrait for
° =0andF = 0 in the Du±ng equation.
To make this portrait, Jennifer inserts the time-dependence of the displacement
by changing the variable x to x(t) in the restoring force,
> f:=subs(x=x(t),f):
and introduces the velocity dx=dt = y(t) ineq0 .
> eq0:=diff(x(t),t)=y(t):
The following three initial conditions are considered.
> ic1:=x(0)=0.09,y(0)=0: ic2:=x(0)=-0.09,y(0)=0:
ic3:=x(0)=-1.5,y(0)=0:
These conditions should produce undamped oscillatory motion in the right,
left, and both potential wells, respectively. To con¯rm this, Jennifer applies
the phaseportrait command to the coupled system eq0 and dy=dt = f.
> phaseportrait([eq0,diff(y(t),t)=f],[x(t),y(t)],t=0..100,
[[ic1],[ic2],[ic3]],stepsize=0.1,,x=-1.5..1.5,color=red,
linecolor=blue,arrows=MEDIUM):
1
y
–1 x 1
–1
Figure 1.13: Phase portrait for inverted Du±ng equation for ° =0andF =0.