Instability in systems is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior.
In control theory, a system is unstable if any of the roots of its characteristic equation has real part greater than zero. This is equivalent to any of the eigenvalues of the state matrix having real part greater than zero.
In structural engineering, a structure can become unstable when excessive load is applied. Beyond a certain threshold, structural deflections magnify stresses, which in turn increases deflections. This can take the form of buckling or crippling. The general field of study is called structural stability.
- Ballooning mode instability (some analogy to the Rayleigh–Taylor instability); found in the magnetosphere
- Atmospheric instability
- Hydrodynamic instability or dynamic instability (atmospheric dynamics)
- Hydrostatic instability or static instability/vertical instability (parcel instability), thermodynamic instability (atmospheric thermodynamics)
- Bénard instability
- Drift mirror instability
- Kelvin–Helmholtz instability (similar, but different from the diocotron instability in plasmas)
- Rayleigh–Taylor instability
- Plateau-Rayleigh instability (similar to the Rayleigh–Taylor instability)
- Richtmyer-Meshkov instability (similar to the Rayleigh–Taylor instability)
Plasma instabilities can be divided into two general groups (1) hydrodynamic instabilities (2) kinetic instabilities. Plasma instabilities are also categorised into different modes:
(azimuthal wave number)
displays harmonic variations of beam radius with distance along the beam axis
|m=1||Sinuous, kink or hose instability:|
represents transverse displacements of the beam crosssection without change in the form or in a beam characteristics other than the position of its center of mass
growth leads towards the breakup of the beam into separate filaments.
|Gives an elliptic cross-section|
|m=3||Gives a pyriform (pear-shaped) cross-section|
Source: Andre Gsponer, "Physics of high-intensity high-energy particle beam propagation in open air and outer-space plasmas" (2004)
List of plasma instabilities
- Shengtai Li, Hui Li "Parallel AMR Code for Compressible MHD or HD Equations" (Los Alamos National Laboratory) 
- Buneman, O., "Instability, Turbulence, and Conductivity in Current-Carrying Plasma" (1958) Physical Review Letters, vol. 1, Issue 1, pp. 8-9
- Kho, T. H.; Lin, A. T., "Cyclotron-Cherenkov and Cherenkov instabilities" (1990) IEEE Transactions on Plasma Science (ISSN 0093-3813), vol. 18, June 1990, p. 513-517
- Finn, J. M.; Kaw, P. K., "Coalescence instability of magnetic islands" (1977) Physics of Fluids, vol. 20, Jan. 1977, p. 72-78. (More citations)
- Uhm, H. S.; Siambis, J. G., "Diocotron instability of a relativistic hollow electron beam" (1979) Physics of Fluids, vol. 22, Dec. 1979, p. 2377-2381.