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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)