Math 2280
Quiz 9
1. Given the system
dx
dt
dy
dt
= εx + y − x(x2 + y 2 )
,
= −x + εy − y(x2 + y 2 )
find the linearization about the fixed point at the origin and classify its stability and type if (a) ε > 0 and (b)
ε < 0.
The Jacobian for the system is
J(x, y) =
ε − 3x2 − y 2
−1 − 2xy
1 − 2xy
ε − x2 − 3y 2
.
We are interested in its eigenvalues at (0, 0), so we must have
ε−λ
1
= (ε − λ)2 + 1 ⇒ (ε − λ)2 = −1 ⇒ ε − λ = ±i ⇒ λ = ε ± i.
0 = |J(0, 0) − λI| = −1 ε − λ
So if ε > 0 the fixed point at (0, 0) is an unstable spiral. If ε < 0, then the fixed point at (0, 0) is an asymptotically
stable spiral.
2. Linearize the system
dx
dt
dy
dt
= 60x − 4x2 − 3xy
= 42y − 2y 2 − 3xy
about the fixed point at (0, 21) and classify the fixed point’s stability and type.
The Jacobian for this system is
J(x, y) =
60 − 8x − 3y
−3y
so
J(0, 21) =
−3x
42 − 4y − 3x
−3
0
−63 −42
,
,
which has eigenvalues λ1 = −3 and λ2 = −42. This tells us that the fixed point at (0, 21) is an asymptotically
stable node.
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