Perturbation of bivariate copulas

New types of constructions of bivariate copulas are introduced, discussed and exemplified. Based on a given copula C  , we look for its perturbation into another copula CHCH, possibly close to C. A special stress is put on the perturbation of the basic copulas M, W, Π. Our results generalize several methods known from the literature, such as the Farlie–Gumbel–Morgenstern copula family, for example. An illustrative example when fitting copulas to real data is also added.