A method for the quantitative analysis of the degree and parameters of synchronization of the chaotic oscillations in two coupled oscillators is proposed, which makes it possible to reveal a change in the structure of attractors. The proposed method is tested on a model system of two unidirectionally coupled logistic maps. It is shown that the method is robust with respect to both the presence of a low-intensity noise and a nonlinear distortion of the analyzed signal. Specific features of a rearranged structure of the attractor of a driven subsystem in the example under consideration have been studied.