Eigenvalue stability
After linearisation, the eigenvalues of the Jacobian determine how small perturbations behave near an equilibrium.
General rule
If every eigenvalue \(\lambda\) of \(J(\mathbf{x}^*)\) has
\[\operatorname{Re}(\lambda)<0,\]the equilibrium is locally asymptotically stable. If at least one eigenvalue has positive real part, the equilibrium is unstable.
| Eigenvalues | Typical local behaviour |
|---|---|
| Both real and negative | Stable node |
| Opposite signs | Saddle; unstable |
| Complex pair with negative real part | Spiral toward equilibrium |
| Complex pair with positive real part | Spiral away |
Borderline cases
If an eigenvalue has zero real part, linearisation alone may be inconclusive and nonlinear terms may need to be examined.
Key idea. Eigenvalues convert the local matrix structure of the model into a classification of how perturbations grow, decay or oscillate.