Positively invariant set

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

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

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