Instability

Instability means that sufficiently small changes of the initial state can eventually produce departures that fail the specified smallness requirement. It concerns the response to perturbations of a selected solution or invariant state, rather than merely the existence of motion. A body at an exact unstable equilibrium can remain there in an ideal mathematical model until it is perturbed.

For $\dot{x}=\lambda x$ with $\lambda >0$, $x(t)=x(0)e^{\lambda t}$ gives exponential separation from zero. A positive growth rate is sufficient for linear instability. Vanishing linear growth is inconclusive: $\dot{x}=x^3$ is also unstable at zero, although its linearization is $\dot{x}=0$. Nonlinear terms can therefore determine the answer near a nonhyperbolic equilibrium. An observed growing signal must also be distinguished from an externally increasing drive, which changes the solution being tested.