Classification of critical points
In section 10.2 of the textbook, the recommended method for doing a problem like #9 is
to first compute the trace and determinant of the matrix A, and then use figure 10.2.12.
This is perfectly correct, but I do not like the method, and think it is more complicated
than the alternative discussed in class. You may safely ignore this technique, and I will
not provide the figure 10.2.12 for you to use on quiz 4 or the final exam.
When asked to classify the critical point (0, 0) your answer should make use of the
following facts concerning the eigenvalues λ1 , λ2 of the nonsingular matrix A.
Stability property:
1. (0, 0) is asymptotically stable if λ1 , λ2 are both negative or if they have the form
α + iβ with α < 0.
2. (0, 0) is unstable if at least one of λ1 , λ2 is positive or if they have the form α + iβ
with α > 0.
3. (0, 0) is stable if λ1 , λ2 = ±iβ.
Shape of solution curves:
1. (0, 0) is a node if λ1 , λ2 are real and of the same sign (Figures 10.2.3, 10.2.4)
2. (0, 0) is a saddle point if λ1 , λ2 are real and of opposite sign. (Figures 10.2.5, 10.2.6)
3. (0, 0) is a spiral point if λ1 , λ2 = α ± iβ with α 6= 0. (Figure 10.2.10)
4. (0, 0) is a center if λ1 , λ2 = ±iβ. (Figure 10.2.9)