Consider a family of planar systems depending on two parameters (n,b) and
having at most one limit cycle. Assume that the limit cycle disappears at some
homoclinic (or heteroclinic) connection when Φ(n,b)=0. We present a method
that allows to obtain a sequence of explicit algebraic lower and upper bounds
for the bifurcation set Φ(n,b)=0. The method is applied to two quadratic
families, one of them is the well-known Bogdanov-Takens system. One of the
results that we obtain for this system is the bifurcation curve for small
values of n, given by b=75n1/2+72/2401n−30024/45294865n3/2−2352961656/11108339166925n2+O(n5/2). We obtain
the new three terms from purely algebraic calculations, without evaluating
Melnikov functions
