New parameterizations of the dynamic conditional correlation (DCC) model and of the regime-switching dynamic correlation (RSDC) model are introduced, such that these models provide a specific dynamics for each correlation. They imply a non-linear autoregressive form of dependence on lagged correlations and are based on properties of the Hadamard exponential matrix. The new models are applied to a data set of twenty stock market indices, comparing them to the classical DCC and RSDC models. The empirical results show that the new models improve their classical versions in terms of several criteria.