Derivatives of the Separation Function of Generalized Saddle Connections

Abstract A classical formula shows that the breaking of a connection between two hyperbolic saddles $$s_0^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> and $$s_0^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> <mml:mo>-</mml:mo> </mml:msubsup> </mml:math> can be studied by means of a convergent improper integral that is often called the Melnikov integral. The goal of this paper is to study the applicability of this formula in more general situations, for instance, when the singularities $$s_0^\pm $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> <mml:mo>±</mml:mo> </mml:msubsup> </mml:math> are semi-hyperbolic or even nilpotent. We will show that 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, Theorem A, 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 non-elementary singularity at infinity. As an application of Theorem A 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.