Lecture Notes in Differential Equations - Bruce E. Shapiro

381
Distinct Real Nonzero Eigenvalues of the Same Sign
If T 2 > 4∆ > 0 both eigenvalues will be real and distinct. Repeated
eigenvalues are excluded because T 2 ≠ 4∆.
IfT > 0, both eigenvalues will both be positive, while if T < 0 both eigenvalues
will be negative (note that T = 0 does not fall into this category). The
solution is given by (34.48); A and B are determined by initial conditions.
The special case A = B = 0 occurs only whenx(t 0 ) = y(t 0 ) = 0, which
gives the isolated critical point at the origin. For nonzero initial conditions,
the solution will be a linear combination of the two eigensolutions
y 1 = ve λ1t (34.50)
y 2 = we λ2t (34.51)
By convention we will choose λ 1 to be the larger of the two eigenvalues
in magnitude; then we will call the directions parallel to v and vthe fast
eigendirection and the slow eigendirection, respectively.
If both eigenvalues are positive, every solution becomes unbounded as t →
∞ (because e λit → ∞ as t → ∞) and approaches the origin as t → −∞
(because e λit → 0 as t → −∞), and the origin is called a source, repellor,
or unstable node.
If both eigenvalues are negative, the situation is reversed: every solution
approaches the origin in positive time, as t → ∞, because e λit → 0 as
t → ∞, and diverges in negative time as t → −∞ (because e λit → ∞ at
t → −∞), and the origin is called a sink, attractor, or stable node.
The names stable node and unstable node arise from the dynamical
systems interpretation: a particle that is displaced an arbitrarily small
distance away from the origin will move back towards the origin if it is a
stable node, and will move further away from the origin if it is an unstable
node.
Despite the fact that the trajectories approach the origin either as t →
∞ or t → −∞, the only trajectory that actually passes through the
origin is the isolated (single point) trajectory at the origin. Thus the only
trajectory that passes through the origin is the one with A = B = 0. To
see this consider the following. For a solution to intersect the origin at a
time t would require
Ave λ1t + Bwe λ2t = 0 (34.52)