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.
Additional recommended knowledge
Fluid instabilities
Fluid instabilities occur in liquids, gases and plasmas, and are often characterized by the shape that form; they are studied in fluid dynamics and magnetohydrodynamics. Fluid instabilities include:
Plasma instabilities
Plasma instabilities can be divided into two general groups (1) hydrodynamic instabilities (2) kinetic instabilities. Plasma instabilities are also categorised into different modes:
Mode (azimuthal wave number)  Note  Description  Radial modes  Description 
m=0   Sausage instability: displays harmonic variations of beam radius with distance along
the beam axis  n=0  Axial hollowing 
n=1  Standard sausaging 
n=2  Axial bunching 
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  
m=2  Filamentation modes: growth leads towards the breakup
of the beam into separate filaments.  Gives an elliptic crosssection 
m=3  Gives a pyriform (pearshaped) crosssection 
Source: Andre Gsponer, "Physics of highintensity highenergy particle beam propagation in open air and outerspace plasmas" (2004)
List of plasma instabilities
 Bennett pinch instability (also called the zpinch instability )
 Beam acoustic instability
 Bumpintail instability
 Buneman instability,^{[2]} (same as FarleyBuneman instability?)
 Cherenkov instability,^{[3]}
 Chute instability
 Coalescence instability,^{[4]}
 Collapse instability
 Counterstreaming instability
 Cyclotron instabilities, including:
 Alfven cyclotron instability
 Electron cyclotron instability
 Electrostatic ion cyclotron Instability
 Ion cyclotron instability
 Magnetoacoustic cyclotron instability
 Proton cyclotron instability
 Nonresonant BeamType cyclotron instability
 Relativistic ion cyclotron instability
 Whistler cyclotron instability
 Diocotron instability,^{[5]} (similar to the KelvinHelmholtz fluid instability).
 Disruptive instability (in tokamaks)
 Double emission instability
 Drift wave instability
 Edgelocalised modes [2]
 FarleyBuneman instability
 Fan instability
 Filamentation instability
 Firehose instability (also called Hose instability)

 Flute instability
 Free electron maser instability
 Gyrotron instability
 Helical instability (helix instability)
 Helical kink instability
 Hose instability (also called Firehose instability)
 Interchange instability
 Ion beam instability
 Kink instability
 Lower hybrid (drift) instability (in the Critical ionization velocity mechanism)
 Magnetic drift instability
 Magnetic buoyancy instability (Parker instability)
 Modulation instability
 NonAbelian instability (see also ChromoWeibel Instability)
 ChromoWeibel Instability
 Nonlinear coalescence instability
 Oscillating two stream instability, see two stream instability
 Pair instability
 Parker instability (magnetic buoyancy instability)
 Peratt instability (stacked toroids)
 Pinch instability
 Sausage instability
 Slow Drift Instability
 Tearing mode instability
 Two stream instability
 Weak beam instability
 Weibel instability
 zpinch instability, also called Bennett pinch instability

Notes
 ^ Shengtai Li, Hui Li "Parallel AMR Code for Compressible MHD or HD Equations" (Los Alamos National Laboratory) [1]
 ^ Buneman, O., "Instability, Turbulence, and Conductivity in CurrentCarrying Plasma" (1958) Physical Review Letters, vol. 1, Issue 1, pp. 89
 ^ Kho, T. H.; Lin, A. T., "CyclotronCherenkov and Cherenkov instabilities" (1990) IEEE Transactions on Plasma Science (ISSN 00933813), vol. 18, June 1990, p. 513517
 ^ Finn, J. M.; Kaw, P. K., "Coalescence instability of magnetic islands" (1977) Physics of Fluids, vol. 20, Jan. 1977, p. 7278. (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. 23772381.
