Lecture Notes in Differential Equations - Bruce E. Shapiro

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Figure 34.5: Phase portraits typical of an unstable star node (left) and an
unstable degenerate node (right). The corresponding stable nodes have the
arrows pointing so that the solutions approach the origin in positive time.
Real Eigenvalues with Opposite Signs
If ∆ < 0, one eigenvalue will be positive and one eigenvalue will be negative,
regardless of the value of T. Denote them as λ and −µ, where λ > 0 and
µ > 0. The solution is
y = Ave λt + Bwe −µt (34.63)
Solutions that start on the line through the origin with direction v (A ≠ 0
but B = 0) diverge as t → ∞ and approach the origin as t → −∞; the
corresponding trajectory is called the stable manifold of the critical
point.
Solutions that start on the line through the origin with direction w ((A = 0
with B ≠ 0) diverge as t → −∞ and approach the origin as t → ∞; the
corresponding trajectory is called the unstable manifold of the critical
point. Besides the stable manifold and the unstable manifold, no other
trajectories approach the origin. The critical point itself is called a saddle
point or saddle node (see figure 34.6).
Example 34.4. The system
}
x ′ = 4x + y
y ′ = 11x − 6y
(34.64)