Computer Algebra Recipes

50 CHAPTER 2. PHASE-PLANE ANALYSIS
For q>0, now consider the case p =0. Thetworoots,¸1 and ¸2, arenow
purely imaginary, viz., ¸1;2 = § iq, and the general solution is of the undamped
oscillatory form u = A cos(qt)+B sin(qt). The solution is characteristic of
trajectories in the vicinity of a vortex point.
Finally, we examine the situation in which q0 has been labeled as
vortices and focal points, rather than vortices alone. This is because the analysis
for the vortices is not de¯nitive for nonlinear models, 2 sincewehavekept
only ¯rst-order terms in u and v in the Taylor expansion (2.4). Higher-order
terms in the expansion may turn vortices into focal points. With the Taylor expansion
option available in Maple, we could, of course, keep higher-order terms
in an attempt to distinguish between the two types of ¯xed points. This can be
done for individual cases, but it is di±cult to make \global" statements that
2 For linear ODE models, recall that higher-order terms are not present and one can have
vortices only for p =0andq>0.
q