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Let $x' = f(x)$ be autonomous first–order equation differential with an equiliburiium point $x_0$.

Suppose $f'(x_0) = 0$ what can I say about the behavior of soluton near $x_0$?

If $f'(x_0) ≠ 0$ and $f''(x_0) = 0$ then what is the dynamical behavior near this point. And identically I have above question for this $f'(x_0) ≠ 0$ and $f''(x_0) ≠ 0$, but $f'''(x_0) ≠ 0$.

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    What are your own thoughts...2012-12-23
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    All-caps is interpreted as shouting on the internet - please don't do it again.2012-12-23
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    Why not make up some examples that are simple enough to solve and see what kind of behavior you can get?2012-12-23

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