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Using polar coordinates, create a system with a fixed point at the origin, and with infinitely many periodic orbits which alternate between clockwise and counterclockwise flows.

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You might want to edit the title of this question. – Jesse Madnick Apr 5 '11 at 7:33
Why did you delete the body of the question? – Rahul Narain Apr 6 '11 at 6:15
in general it is vastly preferable that we keep the contents of question statements as self-contained on this website as possible. Therefore I reverted your edit which replaced the above question Text with a Linked Image. Your edits to this question may be interpreted as borderline vandalism, please refrain from that in the future. – Willie Wong♦ Apr 6 '11 at 12:54
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Please do not ask questions in imperative. – Q__ Apr 6 '11 at 13:08

closed as too localized by Andres Caicedo, Rahul Narain, J. M., Akhil Mathew Apr 6 '11 at 15:28

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2 Answers

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$re^{i\theta} \longrightarrow re^{i(\theta+ (-1)^{[\frac{r}{2 \pi}]}r)} $

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start with circles $(r\cos\theta,r\sin\theta)\mapsto(-\sin\theta,\cos\theta)$. now vary the magnitude, say by $\sin r$: $$ (r\cos\theta,r\sin\theta)\mapsto(\sin r)(-\sin\theta,\cos\theta) $$

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