instability

Some resting states are like a pencil balanced on its tip: in principle it can stand there, but the faintest breath sends it toppling and it never comes back. An equilibrium is unstable when arbitrarily small disturbances can grow — there is no starting ball, however tiny, that keeps every nearby trajectory close. Instability is simply the failure of Lyapunov stability.

Stated as the negation of the definition: an equilibrium x* is unstable if there is some tolerance epsilon such that, no matter how small a starting tolerance delta you pick, at least one trajectory beginning within delta of x* eventually leaves the epsilon-ball around x*. In words, you cannot keep all nearby motions close — at least one always escapes the tube, however close to the equilibrium you start. For a hyperbolic equilibrium this happens exactly when the Jacobian has any eigenvalue with positive real part (a source or a saddle): that direction expands, amplifying the tiniest deviation.

Instability is not a defect to be merely avoided — it is the engine of much interesting behaviour. The unstable equilibrium of a buckling beam, the tipping point in a climate model, the threshold of an epidemic, all hinge on it. There is also a Lyapunov-style test for instability (Chetaev's theorem): if you can find a function that is positive somewhere arbitrarily close to x* and whose rate is positive there, trajectories are pushed away and the equilibrium is unstable. Note a saddle is unstable even though some special trajectories do approach it along the stable direction — almost all nearby ones still leave.