| Abstract: |
| A classical formula shows that the breaking of a connection between two hyperbolic saddles $s_0^+$ and $s_0^-$ can be studied by means of a convergent improper integral that is often called the Melnikov integral. In this talk we will show that this formula extends to more general situations, for instance, when the singularities $s_0^\pm$ are semi-hyperbolic or even nilpotent. In some of these cases the improper integral is no longer convergent but nevertheless, under convenient hypothesis, there is a kind of residue that provides the desired information. Our main result expands the scope of situations in which we can study the breaking of homoclinic or heteroclinic connections. We show that this is indeed the case by analysing three different examples: a heteroclinic connection between nodes, a heteroclinic connection between semi-hyperbolic saddles at infinity and a homoclinic connection in a
singularity at infinity. As an application we obtain a general result aimed at studying the breaking of hemicycles and we present several results to analyse the perturbation of unbounded polycycles within a quadratic unfolding that is versal. |
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