Periodic and almost periodic solutions of the Riccati equations with linear reflecting function

The method of Mironenko’s reflecting function is used for investigation of Riccati equations. The class of Riccati equations with certain-type reflecting function has been preliminarily constructed. The necessary and sufficient conditions, under which the Riccati equation would have a reflecting function linear in phase variable, are proved. These conditions are constructive in nature, since on their basis the formula is obtained, which shows the linear in phase variable reflecting function in terms of the coefficients of the Riccati equation. Additionally, the relationship between the parity (oddness) property of the coefficients of the Riccati equation and the existence of a reflecting function linear in phase variable is investigated. The application of the method of Mironenko’s reflecting function to the constructed class of Riccati equations revealed sufficient conditions, under which all its solutions are periodic or almost periodic. A sign of no periodic solutions for almost periodic Riccati equations is obtained. An example of the quasi-periodic Riccati equation with quasi-periodic reflecting function, which has a periodic solution, is given.