Lecture Notes in Differential Equations - Bruce E. Shapiro

392 LESSON 34. CRITICAL POINTS
Figure 34.9: Phase portraits of singular linear system where all coefficients
are nonzero for c/a > 0.
is a critical point (when d ≠ 0), or on the line x = 0 (when d = 0). The
directions of motion along the trajectories switch along the critical line. A
good analogy is to think of the critical line as the top of a ridge (or the
bottom of a valley), compared to a single apex for a nonsingular system
(Figure 34.9). We essentially have a whole line of sources or sinks.
By a similar argument, if c = d = 0 and a ≠ 0 and/or b ≠ 0, the system
becomes
}
x ′ = ax + by
y ′ (34.102)
= 0
The solutions are all horizontal lines, and there is a ridge or valley of critical
points along the line y = ax/b (if b ≠ 0) or along the line x = 0 (if b = 0).
If a = c = 0 with b ≠ 0 and d ≠ 0, the system is
}
x ′ = by
y ′ = dy
(34.103)
The x-axis (the line y = 0) is a critical ridge (valley) and the trajectories
are parallel lines with slope d/b. Similarly, if b = d = 0 with a ≠ 0 and
d ≠ 0 the system is
}
x ′ = ax
y ′ (34.104)
= ay
so that the trajectories are parallel lines with slope c/a and the y-axis is a