Stability of Linear Systems
Theorem 1. Assume the origin (0, 0) is the only critical point for the linear system
x'(t) = ax + by
y'(t) = cx + dy, or [x'] = [a b][x]
[y'] [c d][y],
where a, b, c, and d are real, and let r1 and r2 be the eigenvalues of the coefficient matrix or, equivalently, the roots of the characteristic equation
r^2 - (a + d)r + (ad - bc) = 0.
The stability of the origin and the classification of the origin as a critical point depends on the roots r1 and r2 as follows:
Roots (eigenvalues) | Type of Critical Point | Stability
distinct, positive | improper node | unstable
distinct, negative | improper node | asymptotically stable
opposite signs | saddle point | unstable
equal, positive | proper or improper node | unstable
equal, negative | proper or improper node | asymptotically stable
complex-valued:
positive real part | spiral point | unstable
negative real part | spiral point | asymptotically stable
pure imaginary | center | stable