For a dynamical system $\dot{x} = f(x)$, I understand the Poincare map is defined by successive intersections of an (n-1) dimensional surface $\Sigma$ with trajectories in n dimensional phase space.
I can see that a fixed point must correspond to a closed orbit (because trajectories in phase space cannot intersect). Is there also a simple interpretation for how a n-cycle of the Poincare map corresponds to orbits?
I can imagine something like a figure 8 orbit would give us a three cycle however I know this is not possible as trajectories cannot intersect! I cannot see anyother way in which a trajectory could intersect 3 (or more) times with $\Sigma$.
In simple terms n-cycle corresponds to a periodic orbit as well. It is easy to see because n-cycle gives you n fixed points of n-th power of the Poincare map.
Pictures like this are very common (here you have 2-cycle)