Abstract

We study the shape of the boundaries of the tongues, and the behaviour close to their tips, for the family of double standard maps $f_{a,b}(x)=2x+a+(b/pi)sin(2pi x)$ (mod 1). It turns out that the shape is fairly regular, mainly due to the realanalyticity of the maps. We present a new method to classify non-generic cusp bifurcations for two-parameter families of one-dimensional systems. We illustrate this method in the generic case.