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).