San José State University
Math 133A, Fall 2005
Solutions to graded Homework 12 problems
Ex. 3.7, #2. (a) The trace is T = a and the determinant D = 2. Therefore, the corresponding curve
C in the trace-determinant plane is the straight line parallel to the T -axis passing through the
point (0, 2).
2
(b) Where does C intersect
√ the critical loci? C intersects the repeated root parabola T = 4D
2
if a = 4 · 2, i.e., if a = ±2 2. It intersects the D-axis when a = 0. Therefore (see Fig. 1):
√
• for a < −2 2, we have a real sink with distinct eigenvalues;
√
• for a = −2 2, a real sink with a repeated eigenvalue;
√
• for −2 2 < a < 0, a (counterclockwise) spiral sink;
• for a = 0, a (counterclockwise) center;
√
• for 0 < a < 2 2, a (counterclockwise) spiral source;
√
• for a = 2 2, a real source with a repeated eigenvalue;
√
• for a > 2 2, a real source.
√
(c) The bifurcation values are therefore a = ±2 2 and a = 0.
D
2
T
√
−2 2
0
√
2 2
Figure 1: Exercise 3.7, #2.
√
Ex. 3.7, #5. (a) The trace is T = a and the determinant D = − 1 − a2 . The corresponding curve
C is the lower semicircle in the trace-determinant plane (Fig. 2).
(b) When a 6= ±1, then D < 0, so the phase portrait is a saddle. If a = −1, then T = −1 and
D = 0, so the eigenvalues are 0 and −1; the phase portrait has a line of equilibria and all other
solutions are straight lines converging to the line of equilibria. If a = 1, then T = 1 and D = 0,
so the eigenvalues are 0 and 1; the phase portrait has a line of equilibria and all other solutions
are straight lines diverging from the line of equilibria.
(c) Therefore, the bifurcation values are a = ±1.
1
D
−1
0
1
T
Figure 2: Exercise 3.7, #5.
Ex. 3.7, #10. The trace is T = 2a and the determinant D = a2 + b2 . Since (2a)2 ≤ 4(a2 + b2 ), we
have T 2 ≤ 4D, for all a, b. Thus for every a, b, the given system falls on or above the repeated
root parabola. Therefore, we always get a spiral sink (a < 0), a spiral source (a > 0), or a center
(a = 0), when the system falls above the repeated root parabola, which is when b 6= 0. If b = 0,
then the eigenvalues are a, a, so we have a real sink when a < 0 or a real source when a > 0
(Fig. 3).
b
spiral sink
spiral source
a
spiral sink
spiral source
Figure 3: Exercise 3.7, #10.
Ex. 3.7, #11. (a) The corresponding one-parameter family of linear systems is
dY
0
1
=
Y.
−3 −b
dt
(b) The trace is T = −b and the determinant D = 3. Since b ≥ 0, the corresponding curve
C in the trace-determinant plane is the straight line ray parallel to the T -axis, starting from
the point (0, 3) and emanating to the right. Note that as b increases, the corresponding point
(T, D) = (−b, 3) on C moves from right to left.
2
(c) Where does C intersect the critical
√ loci? It intersects the repeated root parabola T = 4D
2
when (−b) = 4 · 3, i.e., when b = 2 3. C intersects the D-axis when T = −b = 0. Therefore
(Fig. 4):
2
• for b = 0, we have a (clockwise) center;
√
• for 0 < b < 2 3, a (clockwise) spiral sink;
√
• for b = 2 3, a real sink with a repeated eigenvalue;
√
• for b > 2 3, a real sink with distinct eigenvalues.
D
3
T
√
b=2 3
0
Figure 4: Exercise 3.7, #11.
3