Lecture Notes in Differential Equations - Bruce E. Shapiro

388 LESSON 34. CRITICAL POINTS
Figure 34.7: An unstable spiral node.
which form a complex conjugate pair with negative real part. Hence the
origin is a stable spiral center.
Example 34.6. The system
}
x ′ = ax − y
y ′ = x + ay
(34.77)
where a is a small real number, has a spiral center at the origin. It is easily
verified that the eigenvalues are λ = a ± i. To get an explicit formula for
the spiral we use the following identities:
rr ′ = xx ′ + yy ′ (34.78)
r 2 θ ′ = xy ′ − yx ′ (34.79)
to convert the system into polar coordinates. The radial variation is
rr ′ = xx ′ + yy ′ (34.80)
= x(ax − y) + y(x + ay) (34.81)
= a(x 2 + y 2 ) (34.82)
= ar 2 (34.83)
and therefore (canceling a common factor of r from both sides of the equation),
r ′ = ar (34.84)