Lecture Notes in Differential Equations - Bruce E. Shapiro

386 LESSON 34. CRITICAL POINTS
Figure 34.6: Topology of a saddle point (left) and phase portrait for example
34.4 (right).
stable manifold
saddle
point
unstable manifold
has trace
and determinant
T = 4 − 6 = −2 (34.65)
∆ = (4)(−6) − (1)(11) = −24 + −11 = −35 (34.66)
Thus √
T
2
− 4∆ = √ (−2) 2 − 4(−35) = √ 144 = 12 (34.67)
and the eigenvalues are
λ = 1 2
(
T ± √ )
T 2 −2 ± 12
− 4∆ = = 5, −7 (34.68)
2
Since the eigenvalues ( have different ( signs, the origin is a saddle point. Eigenvectors
are (for 5) and (for -7). The stable manifold is the line
1 1
1)
−1)
y = x (corresponding to the negative eigenvalue), and the unstable manifold
is the line y = −11x (corresponding the positive eigenvalue). A phase
portrait is shown in figure 34.6.