Lecture Notes in Differential Equations - Bruce E. Shapiro

389
The angular change is described by
r 2 θ ′ = xy ′ − yx ′ (34.85)
= x(x + ay) − y(ax − y) (34.86)
= x 2 + y 2 (34.87)
= r 2 (34.88)
so that (canceling the common r 2 on both sides of the equation),
θ ′ = 1 (34.89)
Dividing r ′ by θ ′ gives
and hence
dr
dθ = r′
= ar (34.90)
θ
′
r = r 0 e a(θ−θ0) (34.91)
which is a logarithmic spiral with r(t 0 ) = r 0 and θ(t 0 ) = θ 0 .
Purely Imaginary Eigenvalues
If T = 0 and ∆ > 0 the eigenvalues will be a purely imaginary conjugate
pair λ = ±i∆. The solution is
y = v cos ωt + w sin ωt (34.92)
The origin is called a center. Center’s have the unusual property (unusual
compared to the other types of critical points we have discussed thus far)
of being topologically unstable to variations in the equations, as illustrated
by the following example. A system is topologically unstable if any small
change in the system changes the geometry, e.g., the systems changes from
one type of center to another.
Example 34.7. The system
}
x ′ = x + 2y
y ′ = −3x − y
(34.93)
has a trace of T = 0 and determinant of ∆ = 5. Thus
λ = 1 [
T ± √ ]
T
2
2 − 4∆ = ±i √ 5 (34.94)