Where does instability collide?

In control theory, instability means that a feedback system responds to a small disturbance by growing without bound rather than settling back, so a governor hunts ever more wildly instead of holding a steady speed. Maxwell's 1868 On governors opened the mathematical study of this by linking stable regulation to the roots of the system's characteristic equation. The referent is a property of a system's dynamics, decided by whether its characteristic roots lie in the stable region, and it is a different construct from an unstable atmosphere or an unstable ecosystem, sharing only the idea of perturbations that grow.[1]

Unit

Whether perturbations grow or decay (set by the dominant characteristic root)

Recovery means

Not applicable

Measured by

Determined from the eigenvalues or poles of the linearised system; any in the unstable half-plane means instability

In theoretical ecology, instability means that a community does not return to its equilibrium after a disturbance, with populations instead diverging or oscillating away. May's 1972 Will a large complex system be stable? showed, surprisingly, that larger and more richly connected random food webs are less likely to be stable, making instability a property of the structure of species interactions. The referent is the response of a coupled population system to perturbation, read from its interaction matrix, and it is unrelated to a runaway governor or a buoyant storm beyond the shared notion of not returning to balance.[2]

Unit

Whether a disturbed community returns to equilibrium (a complexity threshold)

Recovery means

Not applicable

Measured by

Judged from the eigenvalues of the community interaction (Jacobian) matrix and the complexity measure of species number, connectance, and interaction strength

In meteorology, (static or convective) instability describes a layering of the atmosphere in which an air parcel displaced upward keeps accelerating away from where it started, because it stays warmer and less dense than its surroundings, which is what lets convection and thunderstorms erupt. Holton and Hakim's An introduction to dynamic meteorology sets out the parcel argument comparing the environmental lapse rate with the adiabatic rate that decides this. The referent is a thermodynamic property of the vertical temperature profile, not a system's feedback dynamics or an ecological web, and it is diagnosed from lapse rates and buoyant energy rather than from eigenvalues.[3]

Unit

Whether a displaced air parcel accelerates away (lapse rate versus the adiabatic rate)

Recovery means

Not applicable

Measured by

Diagnosed by comparing the environmental lapse rate to the dry and moist adiabatic lapse rates, and by buoyancy indices such as convective available potential energy