Could somebody quickly provide me with a good parametrization for the homoclinic solution $$\frac{p^2}{2}-\frac{q^2}{2}+\frac{q^3}{3}=0$$ of the system \begin{aligned} \dot{q}&=p\\ \dot{p}&=q-q^2 \end{aligned} I am trying to evaluate Melnikov function of the related perturbed system of ODEs. Thank you for your attention!

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    $\begingroup$ Can't you solve for $p=p(q)$ and use $q$ as the parameter? $\endgroup$ Sep 28, 2012 at 6:35
  • $\begingroup$ I woke up and I had the same idea. Actually it is even better than that. Due to the structure of my perturbation I can eliminate integrals involving $p$ entirely using the relation and just use $q$. You suggested and excellent parametrization! $\endgroup$ Sep 28, 2012 at 15:07


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