# FAQ: If every equilibrium point can be transformed to the origin and then analyzed using a Lyapunov function, how can a system have both stable and unstable equilibrium points?

From FBSwiki

The reason that you want to transform points to the origin is to that you can look at local stability and ignore the higher order (O(3) and above) terms. The actual location of the equilibrium terms still exists in the functions you use to do the analysis, however. For example: Say you have xdot = f(x), with eq. pt. Xe. Then you do the transform: z = x - Xe, so zdot = xdot = f(x), but x = z + Xe, so f(x) = f(z + Xe), so the information about the actual location of the eq. point is still in the functions, and when you do the stability analysis for f(z+Xe) that'll show the stability of the actual equilibrium point you're looking at.

--Haomiao Huang (October 13, 2004)