The authors propose a model of nonlinear interaction of social network agents, which is based on a matrix generalization of the known one-dimensional logistic mapping. Similar models have not been considered previously and have not been investigated numerically. Despite the relative simplicity of the proposed non-linear model, in the course of numerical experiments, various modes of agent behavior were observed. Namely, one of the agents hesitated in making a certain decision, others were able to "agree" (came to a consensus), some agents lost interest in the issue under discussion (their numerical opinion on this issue was zero). Contribution of the authors is also in the fact that the mathematical description of the model has been translated into software algorithms, the parameters have been tuned and the hypotheses have been numerically tested, all of which have been confirmed. For each numerical experiment, we wrote and debugged the program code in the MatLab package.