Positive invariant set

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

Given a dynamical system ${\displaystyle {\dot {x}}=f(x)}$ and trajectory ${\displaystyle x(t,x_{0})}$ where ${\displaystyle x_{0}}$ is the initial point. Let ${\displaystyle {\mathcal {O}}\triangleq \left\lbrace x\in \mathbb {R} ^{n}|\phi (x)=0\right\rbrace }$ where ${\displaystyle \phi }$ 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 ${\displaystyle x(t,x_{0})\in {\mathcal {O}}\ \forall \ t\geq 0}$

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