This paper deals with the parametric dependence of a particularly simple class of one-dimensional circle maps which may be regarded as nonlinear generalizations of a class of piecewise linear, expanding maps. This class of maps are characterized by two parameters: a first parameter which measures the strength of the nonlinearity, and a second one which is a rotation angle. For small values of nonlinearity, the author computes the invariant measure and shows that it has a singular density of first order in the nonlinearity parameter. However, for larger nonlinearity, the rotation can take a nonexpanding region into a expanding one, with the consequence that the attractor may alternate between hyperbolic and nonhyperbolic regions as the intensity of the nonlinearity or the rotation angle are varied.