I consider the following system \begin{align} &\dot x(t) = \frac{1}{y(t)} - x(t)\\ &\dot y(t) = \frac{1}{x(t) - ay(t)}\left(-by(t)^2 + cx(t)y(t) + d\right) \end{align} where $a,b,c,d \in \mathbb R$ are parameters. I'd like to find the normal form in order to study the stability of limit cycles. I have no idea, however, how one would transform the system into the normal form.


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