Lecture Notes in Differential Equations - Bruce E. Shapiro

390 LESSON 34. CRITICAL POINTS
Since the eigenvalues are purely imaginary, the origin is a center.
If we perturb either of the diagonal coefficients, the eigenvalues develop a
real part. For example, the system
}
x ′ = 1.01x + 2y
y ′ (34.95)
= −3x − y
has eigenvalues λ ≈ 0.005 ± 2.23383i, making it an unstable spiral; and the
system
}
x ′ = 0.99x + 2y
y ′ (34.96)
= −3x − y
has eigenvalues λ ≈ −0.005 ± 2.2383i, for a stable spiral.
The magnitude of the real part grows approximately linearly as the perturbation
grows. In general, the perturbed system
}
x ′ = (1 + ε)x + 2y
y ′ (34.97)
= −3x − y
will have eigenvalues
λ = 1 2
[ε ± √ −20 + 4ε + ε 2 ]
= ε 2 ± i√ 5
√
1 − ε 5 − ε2
20
The results for perturbations of ε = ±0.05 are shown in figure 34.8.
(34.98)
(34.99)
Non-isolated Critical Points
When the matrix A in equation (34.36) is singular, the critical points will
not, in general, be isolated, and they will not fall into any of the categories
that we have discussed so far.
Set ∆ = ad−bc = 0, and suppose that all four of a, b, c, and d are nonzero.
Then we can solve for any one of the four coefficients in terms of the others,
e.g., d = bc/a, so that the trajectories are defined by
dy cx + dy cx + (bc/a)y
= = = c dx ax + by ax + by a = d b
(34.100)
The trajectories are parallel lines with slope c/a. There is only one critical