Consider the vector field on the cylinder given by $$\dot \theta=1 \\\dot y=ay$$ Define an appropriate Poincaré map and find a formula for it.
I know how to find the formula for the map, but I do not know how to define it. I think my main misunderstanding is that I don't know how to visualize a graph that depends on $\theta$ and $y$. I think that the map should be some horizontal line on the cyclinder, but I would not know how to begin to define it. Any help would be great, especially as a picture I can wrap my head around!