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

**R**^{n} with
f:

**R**^{n} ->

**R**^{n} a

Lipschitz continuous vector field on

**R**^{n}. Let V :

**R**^{n} ->

**R** be a

Ljapunov function of the above

ODE. Let M be the

subset of

**R**^{n} defined by

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

where the brackets denote the

standard inner product on

**R**^{n}.

Then any

omega-limit set is a subset of M.

If use an open subset S of **R**^{n} 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).