Lecture Notes in Differential Equations - Bruce E. Shapiro

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Figure 34.8: Topological instability of center nodes. Solutions to equations
(34.97) are plotted for epsilon=0.05, -0.05, -.25, 0.5, with initial conditions
of x, y = 1, 0 (black dot). The bounding box is [−2, 2]×[−2, 2] in each case.
point, at the origin as usual. The nullclines are y = −bx/a (for x) and
y = −(c/d)x for y (figure 34.9). At the other extreme, if all the coefficients
are zero, then every point in the plane is a critical point and there are
no trajectories – wherever you start, you will stay there for all time. If
precisely one of a,b,c or d is zero, but the others are nonzero, the matrix
will be nonsingular so the only remaining cases to consider have one or two
coefficients nonzero.
If a = b = 0 and c ≠ 0 and/or d ≠ 0, we the system becomes
}
x ′ = 0
y ′ = cx + dy
(34.101)
so the solutions are all vertical lines, and every point on the line y = −cx/d