Yes. The Poicare-Bendixon Theorem says that for a dynamical system defined in an open subset of $\mathbb R^2$, every compact, nonempty $\omega$ limit set of an orbit is either
- a periodic orbit
- a fixed point
- a countable number of homoclinic orbits connecting finitely many fixed points.
You found an open subset of $\mathbb R^2$ which is forward invariant with respect to your system, so the system is defined on an open subset of $\mathbb R^2$. This particular subset is important because it contains no fixed points. Therefore, the second and third options are out; you must have a periodic orbit. Since this set is bounded, every $\omega$ limit set is compact and nonempty.
The link you posted is about time-invariance of the dynamical system, meaning that the behavior of the system at some time $t$ is the same as at some other time $t′$.
For instance, the dynamical system $\dot{x}(t)=ax(t)$ where $a\in\mathbb{R}$ is time invariant. If you start the system at time $t_0$ from the state $x(t_0)=x_0$ and if you start the system at time $t_1\ne t_0$ from the state $x(t_1)=x_0$, then the two trajectories will be translated versions of each other.
The dynamical system $\dot{x}(t)=tx(t)$ is not time-invariant, but time-varying. In this case, the initial time will matter.
The composition formula
$${\displaystyle \Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1},x),}$$
works for both time-varying and time-invariant systems (with no input).
Finally, the example you gave is indeed a dynamical system whose state is given by $x(t)=x(t_0)+(t-t_0)^2/2$. In this case, the system has an input and this is not captured by the $\Phi$ formula you gave which is for systems with no input. In fact, the solution to that system is given by
$$x(t)=\Psi(t,s)x(s)+\int_s^t\Psi(t,\theta)\theta d\theta,$$
where $\Phi(t,s,x)=\Psi(t,s)x=x$, $t\ge s$, and where $\Psi(t,s)$ is the so-called state transition matrix (see e.g. https://en.wikipedia.org/wiki/State-transition_matrix). Finally, since the system is time-varying, the trajectory will not only depend on the initial condition but also on the initial time.
If you want to know more about dynamical systems, you may look at "Nonlinear Dynamics and Chaos" by Steven Strogatz. It is usually a good starting point.
Best Answer
It's called a forward invariant set or positively invariant set.
(The condition is that solutions starting in the set stay in the set for all $t \ge 0$. This includes actually being defined for all $t\ge 0$, in contrast to blowing up after finite time; if the set is compact, this detail is nothing to worry about.)