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 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:
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 

