Deriving bar instability mode- formal definition of instability?

[thealyosha]

I am doing an undergraduate project on bars and I am trying to derive the bar instability mode given by Mo et al. It says "whether or not a disk is globally stable depends on the global properties of the disk... it is not possible to write down a universal dispersion relation or stability criterion."

I am wondering what the formal, physical definition of an instability is. I know it has something to do with potential energies. Could someone provide a general definition of an instability? Links to sources would be appreciate, as I might want to cite a definition.

 

[lippyka]

An instability is a process through which a system loses its equilibrium and enters an unstable state. This can occur due to an external force, or the internal dynamics of the system. In general, an instability occurs when the potential energy of the system exceeds the kinetic energy, leading to an exponential increase in the amplitude of the oscillations. Examples of instabilities include Rayleigh-Taylor, Kelvin-Helmholtz, and Jeans instabilities. Sources: 1. https://www.sciencedirect.com/topics/engineering/instability2. https://arxiv.org/pdf/1709.00946.pdf3. https://en.wikipedia.org/wiki/Instability