Computer Algebra Recipes

1.1. PHASE-PLANE PORTRAITS 27
x(t)
1
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
10 20 30 40 50
t
Figure 1.9: x versus t for ¯ =0:2.
Vectoria leaves it as a problem for you to obtain the analytic solution for
the critically damped case.
Unfortunately, Vectoria must leave us for now, since her cell phone has just
rung. Mike's plane has just landed and she's o® to the Metropolis International
Airport to pick him up after he clears immigration and customs.
PROBLEMS:
Problem 1-4: Critical damping
Given ! =1,what¯value corresponds to critical damping of the SHO? Make
a phase-plane portrait for this case using the DEplot command and the initial
condition x(0) = 1, _x(0) = 0. Use this command to plot x(t). Then obtain the
analytic solution.
Problem 1-5: Competition for the same food supply
Two biological species competing for the same food supply are described by the
following nonlinear population number equations:
_
N1 =(4¡ 0:0002 N1 ¡ 0:0004 N2) N1;
_
N2 =(2¡ 0:00015 N1 ¡ 0:00005 N2) N2:
(a) Locate all the stationary points.
(b) Create a tangent ¯eld plot that includes all the stationary points. Identify
thenatureofthesepoints.
(c) Create a phase-plane portrait that includes several representative trajectories
that support your identi¯cation of the stationary points.
(d) Use the scene option to plot N1(t) andN2(t).
(e) Attempt to obtain an analytic solution of the ODE system.