  We illustrate with several new applications the power and elegance of the
Bendixson Dulac theorem to obtain upper bounds of the number of limit cycles
for several families of planar vector fields. In some cases we propose to use a
function related with the curvature of the orbits of the vector field as a
Dulac function. We get some general results for Lienard type equations and for
rigid planar systems. We also present a remarkable phenomenon: for each integer
m greater than one, we provide a simple one parametric differential system for
which we prove that it has limit cycles only for the values of the parameter in
a subset of an interval that decreases exponentially when m grows. One of the
strengths of the results presented in this work is that although they are
obtained with simple calculations, that can be easily checked by hand, they
improve and extend previous studies. Another one is that, for certain systems,
it is possible to reduce the question of the number of limit cycles to the
study of the shape of a planar curve and the sign of an associated function in
one or two variables.
