978-3-319-17852-3

182 4. Linear Systems
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Figure 4.3 The vector field indicating a clockwise, elliptical flow of the system
x ′ = y, y ′ = −4x around the origin. The x-nullcline is the x axis (y =0)where
the vector field is vertical, and the y-nullcline is the y axis (x =0)wherethe
vector field is horizontal.
Example 4.3
For the system
x ′ = y,
y ′ = −4x,
the vector field at the point (x, y) isgivenby
F =(y, −4x).
Figure 4.3 shows the vector field, computed using software, at a set of points in
the xy plane. Easily we can visualize the the rotational nature of the orbits. It is
shown in the next example that the orbits are a family of ellipses x 2 + 1 4 y2 = C
that fit into this field. Without software we can make the following argument.
If y>0 (upper half plane), then x ′ > 0 and therefore x must be increasing; if
y<0 (lower half plane) then x ′ < 0andx must be decreasing. If x>0(right
half plane), then y ′ < 0, so y must be decreasing; if x>0 (left half plane), then
y ′ > 0, so y must be increasing. Observe also that the vector field is vertical
(up or down) along x ′ = y = 0, which is along the x axis; the x axis is the
x-nullcline. The vector field is horizontal (right or left) along y ′ = −4x =0,
which is the y axis; the y axis is the x-nullcline. This is all consistent with the
direction field in Figure 4.3. Note that even though a vector field for a system
has a rotational nature, its orbits may not be closed paths, like ellipses. Orbits
that spiral around and into the origin could equally fit the field. Therefore the
vector field itself does not tell the whole story, but it can supplement analytical
calculations to get fine structure properties of orbits. □