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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.

PhiEarl
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  • I do not understand anything you just wrote. Can you ask a more precise question? What does it mean to "prescribe infinity as periodic orbit"? Periodic orbit of what? "If your trajectory is chaotic" - what trajectory? Is this a question about a dynamical system? What kind, on what phase space? – Qiaochu Yuan Jan 08 '21 at 05:09
  • https://en.wikipedia.org/wiki/Riemann_sphere

    Chaos means you can't know the end-result of the trajectory, but if a trajectory converges to infinity, it is then not chaotic, implying boundedness is a needed quality.

    This concept could be extended. It could then imply that no matter how you transform the manifold, a chaotic trajectory will never be periodic, not just in the original Euclidean space, but any space derived from the topology of a differentiable surface.

     – PhiEarl Jan 08 '21 at 20:20
  • I guess that homeomorphic transformations of a plane can stretch a lot of trajectories. Namely, I expect that I can prove the following equivalence. Let $f:\Bbb R\to\Bbb R^2$ be an injective continuous map. Then there exists a homeomorphism $h$ of $\Bbb R^2$ such that $hf(x)=(x,0)$ for each $x\in\Bbb R$ iff $f$ has a continuous extension $\bar f:\Bbb R\cup{\infty}\to \Bbb R^2\cup{\infty}$ such that $\bar f(\infty)=\infty$. – Alex Ravsky Jan 09 '21 at 07:09
  • Here $\Bbb R\cup{\infty}$ and $\Bbb R^2\cup{\infty}$ are Aleksandroff one-point compactifications of spaces $\Bbb R$ and $\Bbb R^2$, respectively. The space $\Bbb R\cup{\infty}$ is homeomorphic to a circle and I guess that the space $\Bbb R^2\cup{\infty}$ is homeomorphic to the Riemann sphere. – Alex Ravsky Jan 09 '21 at 07:09
  • Right, I'm suggesting to extend that concept and to include trajectories, that no matter what smooth homeomorphism one has, there is not way to make a chaotic trajectory periodic. – PhiEarl Jan 09 '21 at 21:56

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