We study a likelihood ratio test for testing the conditional mean of a class of piece-wise stationary CHARN models. We establish the locally asymptotically normal (LAN) structure of the family of likelihoods under study. We prove that the test is asymptotically optimal, and we give an explicit form of its asymptotic local power. We describe an algorithm for detecting change points and estimating their locations. The estimates are obtained as time indices, maximizing the estimate of the local power. The simulation study we conduct shows the good performance of our method on the examples considered. This method is also applied to a set of financial data.