If you map the plane to the Riemann sphere or torus, you can technically prescribe infinity as periodic orbit.
However, if your trajectory is chaotic, not even that case will occur, but what I anticipate more so is that no matter how you transform the manifold, no smooth manifold mapped from $\mathbb{R}^n$ will yield a periodic orbit. Chaotic trajectories are not only chaotic in the solution manifold, but possibly to some sort of extent, any smooth homotopy of it as well.