I am beginning to study dynamical systems. We are given $U \subset \mathbb{R}^n$ open, a vector field $f: U \to \mathbb{R}^n$, and an associated evolution operator for fixed $t \in \mathbb{R}$ $\Phi_t : U \to \mathbb{R}^n$ given by $\Phi_t(x_0) = x(t)$, where $x$ is the solution to the ODE \begin{equation} \dot x = f(x) \\ x(0) = x_0. \end{equation}
We say that $A \subset U$ is invariant if $\Phi_t(A) = A \ \forall t \in \mathbb{R}$. On the other hand, we have definitions for positive (negative) invariance in which $\Phi_t(A) \subset A \ \forall t \geq 0 \ ( \leq 0)$. I am wondering what added conditions (in addition to both positive and negative invariance) would lead to invariance.
It follows from the definitions (those that you detail) that a set $A$ is invariant if and only if it is positively and negatively invariant. The reason is that since $\Phi_t$ is a flow, you have $(\Phi_t)^{-1}=\Phi_{-t}$ and so a set is negatively invariant if and only if $\Phi_t(A)\supset A$ for all $t\ge0$.