Lecture Notes in Differential Equations - Bruce E. Shapiro

378 LESSON 34. CRITICAL POINTS
Figure 34.1: Varieties of isolated critical points of linear systems. The
circles represented neighborhoods; the lines trajectories; the arrows the
direction of change with increasing value of t in a solution.
Sink Source Saddle Center
2. Source, Repellor, or Unstable Node if all such points move away
from P ;
3. Center, if all trajectories loop around P in closed curves;
4. Saddle Point or Saddle Node if some solutions move toward P
and some move away.
Let us begin by studying the linear system
where the matrix
}
x ′ = ax + by
y ′ = cx + dy
( ) a b
A =
c d
(34.35)
(34.36)
is nonsingular (i.e., its determinant is non-zero and the matrix is invertible).
To find the critical points we set the derivatives equal to zero:
0 = ax ∗ + by ∗ (34.37)
0 = cx ∗ + dy ∗ (34.38)
The only solution is (x ∗ , y ∗ ) = (0, 0). Hence (34.35) has a single isolated
critical point at the origin.
The solution depends on the roots of the characteristic equation, or eigen-