It is found that a fixed point of the renormalization group equation corresponding to a system of a unimodal map with extremum of power κ and a map summarizing values of a function of the dynamical variable of the first subsystem, undergoes a bifurcation in the course of increase of κ. It occurs at κ _c =1.984396 and results in a birth of the period-2 stationary solution of the RG equation. At κ=2 this period-2 solution corresponds to the universal period-doubling behavior discovered earlier and denoted as the C-type criticality (Kuznetsov and Sataev in Phys. Lett. A 162:236–242, 1992). By combination of analytical methods and numerical computations we obtain and analyze an asymptotic expansion of the period-2 solution in powers of Δκ=κ−κ _c . The developed approach resembles the ε-expansion in the phase transition theory, in which a “trivial” stationary point of the RG transformation undergoes a bifurcation that gives rise to a new fixed point responsible for the critical behavior with nontrivial critical indices.