Computer Algebra Recipes

20 CHAPTER 1. PHASE-PLANE PORTRAITS
J
1
0.5
–1 –0.5 0.5 1
R
–0.5
–1
Figure 1.3: Tangent ¯eld for Romeo and Juliet's love a®air.
The tangent ¯eld for Romeo and Juliet's love a®air is shown in Figure 1.3. Inspecting
the graph and comparing with Figure 1.1, it is seen that the stationary
point at the origin is a saddle point. By changing the coe±cient values, the
nature of the stationary point and therefore the nature of the love a®air can be
altered. This is left as a problem at the end of the section for you to explore.
The four initial conditions are now entered,
> ic1:=(R(0)=-0.25,J(0)=1): ic2:=(R(0)=-0.27,J(0)=1):
ic3:=(R(0)=0.27,J(0)=-1): ic4:=(R(0)=0.25,J(0)=-1):
and the phaseportrait command is used to create the phase-plane portrait for
the four initial conditions. These conditions are entered as a list of lists. The
time step size is taken to be 0.05. The default is to divide the time range into 20
equal steps. So the default step size here would be 4=20 = 0:2. Again MEDIUM
arrows are chosen, which are colored red. The trajectories are colored blue using
the linecolor option. A complete list of options may be found under DEplot,
whose Help page may be accessed though the topic search.
> phaseportrait([de1,de2],[R(t),J(t)],t=0..4,[[ic1],[ic2],
[ic3],[ic4]],stepsize=0.05,dirgrid=[30,30],R=-1..1,J=-1..1,
arrows=MEDIUM,color=red,linecolor=blue);
The phase-plane portrait for Romeo and Juliet's love a®air is reproduced in
Figure 1.4. Asymptotically, the trajectories approach the separatrixes of the
saddle point at the origin, the separatrixes dividing the phase plane into four
di®erent \°ow" regions for the tangent arrows. Thus, for example, one can see
that for any initial condition in the lower right region, R will remain positive
and J negative. Romeo's love for Juliet is unrequited!