Approximate boundaries of resonance tongue for planar maps

A resonance tongue of a planar map with a weakly resonant equilibrium is a region of parameters in which periodic orbits appear on the invariant circle arising from the Neimark-Sacker bifurcation. Although conditions were given for the existence of resonance tongues, it is still difficult to compute boundaries of such a tongue. Known results give the common term of the lowest order and the different term of the lowest order only for the tongue with a mistake. In this paper we give a method to compute approximately all common terms and the first different term for the boundaries and correct the mistake for the normal form of planar maps having a weakly resonant equilibrium. Then, computing higher order derivatives of implicit functions and discussing Holder smoothness of implicit functions, we use the inducing relation of parameters to compute approximately boundaries of resonance tongues for given planar maps. Finally, we apply our result to a planar discrete time prey-predator model.