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My question concerns an article by Koiller and Carvalho found here: http://link.springer.com/article/10.1007/BF01260390

On page 645, they parameterize the time variable $t$ in terms of one of the phase space "coordinates" (i.e. functions of time $p_1 \colon R \to R$). I understand that a reader can reconstruct $t$ from $p_1$ since $p_1$ is injective on some domain, but I am having a hard time seeing how the authors can replace implicit time dependence with explicit time dependence. (In analogy, just because the solution to a Hamiltonian implies $p = \omega t$ does not mean $\omega \partial_p H = \partial_t H)$.

I am particularly troubled by the fact that the argument found here seems to suggest three vortices in a plane might be non-integrable. In particular, as the authors mention on page 649, all terms are odd when $t_0 = 0$ and clearly both terms survive when $t_0 \neq 0$ on page 651. (They are even in danger of divergence).

Can anyone explain why the authors can trade implicit time dependence for explicit time dependence in the Melnikov formula? An exact answer might prove or disprove that, for a Hamiltonian with no explicit time dependence, if one of the coordinates (say $p$) is injective along the separatrix then a simple zero of the function $$M(t_0) = \int_{-\infty}^{+\infty}\{F, H\}(q(t), p(t+t_0))dt$$ implies chaos where $F$ and $H$ are the perturbation and non-perturbation Hamiltonians, and the integral is along the separatrix.

Thanks for your help.

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