How to show that all trajectories of the system approach a unique periodic solution? Consider an overdamped linear oscillator forced by a square wave.
The system can be nondimensionalized to $\dot{x}+x=F(t)$, where $F(t)$
is a square wave of period $T$:
$F(t)=\begin{cases}
+A, & 0<t<T/2\\
-A, & T/2<t<T
\end{cases}$
where $t\in(0,T)$, and $F(t)$ is periodically repeated for all other
$t$. The goal is to show that all trajectories of the system approach
a unique periodic solution. Now, in order to solve this I am supposed
to use Poincaré map, and the first step in solving this problem is
to show that: 
$x(T)=x_{0}e^{-T}-A(1-e^{-T/2})^{2}$, where $x(0)=x_{0}$.
To be honest I do not even know where to start. Any ideas? How do
I transform this $1$-D time dependent system into a time-independent
system on a cylinder?
 A: 
How do I transform this 1-D time dependent system into a time-independent system on a cylinder?

Well, there is an old trick for that. If you have non-autonomous system $\dot{x} = \mathcal{F}(x, t)$, you can convert it into autonomous system by adding 'fictitious' variable for time and an additional equation:
$$ \frac{dx}{dt} = \mathcal{F}(x, \theta), \,\,\, \frac{d\theta}{dt} = 1 .$$
Now this system is autonomous. In your case this will lead to the system
$$ \dot{x} = -x + F(\theta),\;\;\; \dot{\theta} = 1, $$
and vector field defined by this system is $T$-periodic in $\theta$. So, the phase space is $\mathbb{R} \times \mathbb{S}$ ($\mathbb{R}$ is for $x$ variable and $\mathbb{S}$ for $\theta$ variable) which is cylinder.
There are many ways to solve this equation. @Did already mentioned one, but I suggest to use formula that could be derived from Lagrange's method:
$$ x(t) = e^{-t} \left ( x(0) + \int\limits_{0}^{t} F(u) e^u \, du \right ). $$
(Note that solution will be only piecewise-smooth.)
As it is already stated in your exercise, you need to compute time-$T$ mapping which in this case is just $x(T)$. Periodic solutions have the property that $x(T) = x(0)$, so they are fixed points of time-$T$ mapping. You only need to find fixed points of time-$T$ mapping and study their stability. This will almost finish the proof of what you've been asked in this problem. Here you go.
