Instability of a linear equilibrium

For a constant-matrix linear system $y'=Ay$ , the zero equilibrium is linearly unstable in the dynamical sense if there is $\eta>0$ such that, for every $\delta>0$ , some initial datum with $\|y(0)\|<\delta$ has $\|y(t)\|>\eta$ at a later time. This is failure of Lyapunov stability for this linear equation.

A growing eigenmode

If $Av=\lambda v$ with a real $\lambda>0$ and $v\ne0$ , then $y(t)=c e^{\lambda t}v$ proves instability. A complex eigenvalue with positive real part also gives exponentially growing modes. A nondiagonal Jordan block at an imaginary eigenvalue can cause polynomial growth, so positive exponential growth is a sufficient condition rather than the definition.

Scope for nonlinear equations

Instability of a linearized equation does not by itself prove nonlinear blowup. One must show that a nonlinear solution follows the growing mode for the required interval and control the remainder. Large but bounded transient amplification alone is compatible with Lyapunov stability of a fixed finite-dimensional linear system.