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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.