In the present paper we examine the behavior of a 3-branched linear map which resulted from computations of the periodically perturbed Oregonator. It turns out that the linear map describes quite satisfactorily the period composition predicted by the chemical model and by the more elaborated nonlinear hyperbolic maps that were described in a previous publication. In particular, we have shown that in any interval of the control parameter, two periodic orbits, differing by one point, coexist with one being stable and the other unstable. This phenomenon is, to the best of our knowledge, described for the first time in the literature.