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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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    Please do not ask questions in imperative.2011-04-06

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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) $