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

Author:

Belokursky M. S.1

Affiliation:

1. Francisk Skorina Gomel State University

Abstract

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.

Publisher

Publishing House Belorusskaya Nauka

Subject

General Medicine

Reference15 articles.

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2. Zhou Z. Research on the properties of some planar polynomial differential equations. Applied Mathematics and Computation, 2012, vol. 218, no. 9, pp. 5671–5681. https://doi.org/10.1016/j.amc.2011.11.062

3. Zhou Z. On the structure of the equivalent differential systems and their reflecting integrals. Bulletin of the Brazilian Mathematical Society. New Series, 2017, vol. 48, no. 3, pp. 439–447. https://doi.org/10.1007/s00574-016-0026-4

4. Musafirov E. V. Reflecting function and periodic solutions of differential systems with small parameter. Indian Journal of Mathematics, 2008, vol. 50, no. 1, pp. 63–76.

5. Musafirov E. V. Admissible perturbations of Langford system. Problemy Fiziki, Matematiki i Tekhniki = Problems of Physics, Mathematics and Technics, 2016, no. 3 (28), pp. 47–51 (in Russian).

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