I found this playing with a 2D-ODE-system plotter I'm writing. Surely, since it's so simple, it's been found and extensively studied by someone. What's it called? I'd like to look it up and learn a little about it. There's a weird pattern were some are simple spriograph-like graphs and the others are completely chaotic.
In case anyone enjoys seeing these as much as I do, I'll add an album on imgur for your viewing pleasure. The titles are of the form "$x_0$_$y_0$". These are all rendered with VODE from Scipy with $t_0=0$ and $t_1=1000$.
$$ \begin{eqnarray*} x' &=& \cos(y)+\sin(t) \\ y' &=& \sin(x)+\cos(t) \end{eqnarray*} $$
Here's the album: http://imgur.com/a/lbhrX
If anyone's interested, I uploaded a video of (1,1) plotted from t=0 to 250 with the vector field: https://vimeo.com/88323596.