426 Ordinary Differential Equations
In examining the stability of the critical point at the origin for system (4),dz/ dt = Az, we have assumed that det A # 0. It follows from this assumption
that .A = 0 cannot be an eigenvalue of the matrix A. For if .A = 0 were
an eigenvalue of A, then .A = 0 would, by definition of an eigenvalue, be
a root of the equation det (A - .AI) = 0. But substitution of .A = 0 into
this equation yields the contradiction det A = 0. Consequently, Table 10. l
summarizes the stability properties of the critical point at the origin for the
system (4), dz/dt = Az, in terms of the eigenvalues of A under the assumption
detA # 0.
Table 10.1 Stability Properties of the Linear System dz = Az when
dtdet A# 0
Eigenvalues Stability.A1, .A2 < 0 Asymptotically stable
.A1, .A2 > 0 Unstable
.A1 > 0, .A2 < 0 Unstable
.A= a± i{J ([3 # 0)
()/, < 0 Asymptotically stable
()/, > 0 Unstable
.A= ±i{J ({3 # 0) Neutrally stable
Now let us consider the autonomous system
dxdt = f(x, y)
(1)
dydt = g(x, y)
Type of
critical pointNodeNodeSaddle pointSpiral point
Spiral pointCenterunder the assumption that f(x, y) or g(x, y) is nonlinear. Recall that the
point (x,y) is a critical point of (1), if f(x,y) = g(x,y) = 0. When f
and g have continuous partial derivatives up to order two at (x, y), then we
have from the two dimensional version of Taylor series expa nsion for f and g
about (x, y)
f(x, y) = f(x, y) + fx(x, y)(x - x) + fy(x, y)(y - y) + R1 (x, y)