讲座内容：

An Arnold tongue of a planar map is a region of parameters in which periodic orbits appear on the invariant circle arising from the Hopf (Neimark-Sacker) bifurcation. Although conditions were given for the existence of those tongues, it is still difficult to compute their boundaries of them. We give a method to compute approximately all common terms and the first different term for the boundaries. Then, discussing higher order derivatives and Hölder smoothness for implicit functions, we use the inducing relation of parameters to compute approximately boundaries of tongues for given planar maps and apply the method to a planar discrete time prey–predator model.