  We prove that star-like limit cycles of any planar polynomial system can also
be seen either as solutions defined on a given interval of a new associated
planar non-autonomous polynomial system or as heteroclinic solutions of a
3-dimensional polynomial system. We illustrate these points of view with
several examples. One of the key ideas in our approach is to decompose the
periodic solutions as the sum of two suitable functions. As a first application
we use these new approaches to prove that all star-like reversible limit cycles
are algebraic. As a second application we introduce a function whose zeroes
control the periodic orbits that persist as limit cycles when we perturb a
star-like reversible center. As far as we know this is the first time that this
question is solved in full generality. Somehow, this function plays a similar
role that an Abelian integral for studying perturbations of Hamiltonian
systems.
