Positive invariant set

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

Given a dynamical system \dot{x}=f(x) and trajectory  x(t,x_0) \, where  x_0 \, is the initial point. Let  \mathcal{O} \triangleq \left \lbrace x \in \mathbb{R}^n| \phi (x) = 0 \right \rbrace where \phi 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

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

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