Definition (Invariant Set)

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 $$x_{0}\in \mathcal{O} \implies x(t,x_{0})\in \mathcal{O}\ \forall \ t\geq 0.$$In other words, once a trajectory of the system enters $\mathcal {O}$, it will never leave it again.

Definition (Forward Invariant)

Consider a set $\mathcal{C}$ and initial condition $x(0)=x_{0}$. $\mathcal{C}$ is forward invariant if $$x_{0}\in \mathcal{C} \implies x(t)\in \mathcal{C},\,\forall t\ge 0$$i.e. if we start in $\mathcal{C}$ we stay in $\mathcal{C}$.