San José State University
Math 133A, Spring 2011
Quiz 4 Solutions
Consider the system
dY
a −1
=
Y.
2 0
dt
(a) Sketch the corresponding curve in the trace-determinant plane.
(b) Discuss different types of behavior exhibited by the system as a increases.
(c) Identify the values of a where the type of the system changes.
Solution: (a) The trace of the matrix of the system is T = a and the determinant is
D = 2. The corresponding curve the horizontal straight line L passing through the point
(T, D) = (0, 2), depicted in Fig. 1.
3
2.5
spiral sink
sink
source
spiral source
2
1.5
1
0.5
-4
-3.2
-2.4
-1.6
-0.8
0
0.8
1.6
2.4
3.2
4
-0.5
Figure 1: The trace-determinant plane.
(b) The line L intersects the √
repeated √
root parabola T 2 =√4D at the points (T, D) = (a, 2)
2
where a√ = 8, i.e., when a = ± 8 = ±2 2. When a < −2 2, the system is a sink. When
a = −2 2,
√ the system is a sink with a repeated eigenvalue and only one line of eigenvectors.
When −2 √2 < a < 0, the system is a spiral sink. When a √
= 0, the system is a center. When
0 < a < 2 2, the system is a spiral source. When a = 2 √2, the system is a source with a
repeated eigenvalue and only one eigenvector. When a > 2 2, the system is a source.
√
√
(c) The bifurcation values of a are −2 2, 0, and 2 2.