Dynamical equilibrium does not guarantee dynamical stability. In general, an equilibrium is said to be stable if the system remains bounded, i.e. confined to its neighbourhood, after being subjected to a small perturbation. There are many different classifications of stability, e.g. linear stability implies small perturbations, while non-linear stability allows for perturbations of arbitrary size. The former is equivalent to spectral stability, which occurs when all eigenvalues of the linearized dynamical operator have real parts which are positive or zero. The most general formulation of linear stability is the energy principle, which states that an equilibrium point is stable when it represents a minimum of the potential energy of the system. Most dynamical systems have both stable and unstable equilibria, e.g. the lowest and highest position of a pendulum. Having identified an equilibrium point, one should next investigate its stability properties and, if the point proves unstable, the physical mechanism responsible for the instability.

Hydrodynamic waves and instabilities

By way of introduction to plasma instabilities, let us consider the stability properties of a stratified neutral fluid in the presence of a gravitational field g = gêg.