A new method for finding cycles by semilinear control

In the paper two new results that prove typically real and local univalent properties of certain class of rational functions in the unit disc $\mathbb{D}$ are presented. They provide a theoretical justification of a new method for finding periodic solutions of the discrete vector dynamical systems. The main goal of the paper is to propose and investigate the two-delays generalization of the scheme introduced by Ö. Morgül. The analysis shows that the coefficients in the proposed method can be chosen in a way that improves the admissible range of parameters. The method is elementary and easy to implement. Furthermore, numerical experiments with several well known maps show effectiveness of the control scheme.