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March 31st, 2015, 06:08 AM   #1
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Topology equivalence in dynamical system

Hi, my name is Eric.

I've got trouble when proofing that system
$\displaystyle \dot{x}=\alpha+x^2+O(x^3)$ is topological equivalence with system $\displaystyle \dot{x}=\alpha+x^2$.
I don't understand how to build the homeomorphism for the orbit.
In literature, I read that for $\displaystyle \alpha>0$, the homeomorphism mapping is defined by $\displaystyle h_\alpha(x)=x$, whereas when $\displaystyle \alpha<0$ define $\displaystyle h_\alpha(x)=a(\alpha)+b(\alpha)x$.

What is the function $\displaystyle a(\alpha) \text{ }b(\alpha)$ explicitly?
Could someone please help me?
Thank you so much.
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