Invariance Principle (idea)

The following theorem is called LaSalle's invariance priciple (or sometimes just invariance principle).
Let

d
-- x = f(x)
dt

an autonomous ordinary differential equation in Rⁿ with f: Rⁿ -> Rⁿ a Lipschitz continuous vector field on Rⁿ. Let V : Rⁿ -> R be a Ljapunov function of the above ODE. Let M be the subset of Rⁿ defined by

           / d              \
M := { x | | -- V(x) , f(x) | = 0 }
           \ dx             /

where the brackets denote the standard inner product on Rⁿ.
Then any omega-limit set is a subset of M.

If use an open subset S of Rⁿ instead of the whole space, you'll the same theorem restricted to S (of course you'll have to use S in the definition of M).

Note that limit sets don't have to consist always of equilibrium points. So sometimes a Ljpunov function might indeed provide new information about the dynamics of a system (without knowing the solution).