Let $U\subset \mathbb R^n$ be open, let $f\in C^1(U;\mathbb R^n)$, and let $x^*\in U$ be an equilibrium. If $Jf_{x^*}$ has a complex eigenvalue $\lambda$ with $\operatorname{Re}(\lambda)>0$, then $x^*$ is not Lyapunov stable.
Knowledge Status
Analysis
Discussion
A nonlinear equilibrium is unstable when the linearised dynamics has an eigenvalue whose real part is positive.