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I currently have a definition that states that given a flow $f$, $f$ is structurally stable if for any $g$ in some neighborhood of $f$, $f$ and $g$ are topologically conjugate.

Would the definition of $C^r$ Structural Stability be that for a flow$f$, $f$ is structurally stable if for any $g$ that is $C^r$ close to $f$ (i.e. $f$ and its first $r$ derivatives are within $\epsilon$ of $g$ and its first $r$ derivatives respectively), $f$ and $g$ are topologically conjugate?

I am a little confused on how this is a different definition than the first one. Any elaboration would be appreciated.

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  • $\begingroup$ The first definition coincides with the second one, if by "some neighborhood" you mean a neighborhood in the $C^r$ topology. $\endgroup$ – Hans Lundmark May 11 '15 at 20:01
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Your definitions are right. The difference is that in the first case you have a flow (vector field) that may be only $C^1$-smooth and you are looking at its neighborhood in $C^1$-topology. For the second one your vector field should be at least $C^r$ and the neighbourhood in the $C^r$-topology.

Note that non of them imply the other one.

Note also that usually use define stability up to a time change (unless you include it the definition of the top. conjugacy of flows).

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