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# Instability

[in-stuh-bil-i-tee]

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.

## 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 cross-section 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 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

 Bennett pinch instability (also called the z-pinch instability ) Beam acoustic instability Bump-in-tail instability Buneman instability, (same as Farley-Buneman instability?) Cherenkov instability, Chute instability Coalescence instability, Collapse instability Counter-streaming instability Cyclotron instabilities, including:*Alfven cyclotron instability *Electron cyclotron instability *Electrostatic ion cyclotron Instability *Ion cyclotron instability *Magnetoacoustic cyclotron instability *Proton cyclotron instability *Nonresonant Beam-Type cyclotron instability *Relativistic ion cyclotron instability *Whistler cyclotron instability Diocotron instability, (similar to the Kelvin-Helmholtz fluid instability). Disruptive instability (in tokamaks) Double emission instability Drift wave instability Edge-localised modes Electrothermal instability Farley-Buneman 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 Non-Abelian instability (see also Chromo-Weibel Instability) Chromo-Weibel Instability Non-linear 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 z-pinch instability, also called Bennett pinch instability