Linear Stability Analysis

This analysis helps us in understanding the physics of the problem without solving the complex differential equation governing it.
In this analysis we start with a system which is in equilibrium state. A small perturbation is given to the system and we see whether the system returns to its equilibrium state.

The small perturbation can be written as

where

and

Therefore, we can neglect the higher order terms in and discard all the non-linear terms in . So,