Timeline for Given a vector field all of whose integral curves are closed, is the period a smooth function?
Current License: CC BY-SA 3.0
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| Aug 6, 2012 at 14:30 | comment | added | agt | For example, if $M$ is the Moebius' band $[0,1]\times\mathbb R/\sim$ and $X=\frac{\partial}{\partial x}$ then the period is$$\tau([x,y]_\sim)=\begin {cases}1&\text{if }y=0\\2&\text{if }y\neq 0\end{cases}.$$ | |
| Aug 6, 2012 at 14:25 | comment | added | agt | 1) @Liviu Nicolaescu I have edited the question to point out that the stationary points aren't allowed, i.e. constant solutions aren't considered periodic 2) @Sebastian I have thought what you say, but as with Poincarè map, if the integral curve $\gamma$ starts at $q\in N,$ then its first return on $N$ could be at another point than $q.$ | |
| Aug 6, 2012 at 14:06 | comment | added | Liviu Nicolaescu | What if $X(p)=0$? | |
| Aug 6, 2012 at 9:59 | history | answered | Sebastian | CC BY-SA 3.0 |