Positive invariant set

In mathematical analysis, a positively invariant set is a set with the following properties:

Given a dynamical system x ˙ = f ( x ) and trajectory x ( t , x 0 ) where x 0 is the initial point. Let O ≜ { x ∈ R n | ϕ ( x ) = 0 } where ϕ is a real valued function. The set O is said to be positively invariant if x 0 ∈ O implies that x ( t , x 0 ) ∈ O   ∀   t ≥ 0

Intuitively, this means that once a trajectory of the system enters O , it will never leave it again.