Positively invariant set

In mathematical analysis, a positively (or positive) invariant set is a set with the following properties:

Suppose $$\dot{x}=f(x)$$ is a dynamical system, $$ x(t,x_0) $$ is a trajectory, and $$ x_0 $$ is the initial point. Let $$ \mathcal{O} := \left \lbrace x \in \mathbb{R}^n\mid \varphi (x) = 0 \right \rbrace$$ where $$\varphi$$ is a real-valued function. The set $$\mathcal{O}$$ is said to be positively invariant if $$x_0 \in \mathcal{O}$$ implies that $$x(t,x_0) \in \mathcal{O} \ \forall \ t \ge 0 $$

In other words, once a trajectory of the system enters $$\mathcal{O}$$, it will never leave it again.