# On the existence of a conditionally periodic solution of one quasilinear differential system in the critical case

### Abstract

In the theory of nonlinear oscillations one often encounters conditionally periodic oscillations resulting from the superposition of several oscillations with frequencies incommensurable with each other. When finding a solution to a resonant quasilinear differential system in the form of a conditionally periodic function, the problem of a small denominator arises. Consequently, the proof of the existence, and even more the construction of such a solution is not an easy task. In this article, drawing on the work of VI. Arnold, I. Moser, and other researchers proved the existence and constructed a conditionally periodic solution of a second-order quasilinear differential system in the critical case. Accelerated convergence method by N.N. Bogolyubova, Yu.A. Mitropolsky, A.M. Samoylenko. The result can be applied to construct a conditionally periodic solution of specific differential systems.### References

[1] Kolmogorov A.N., "On the conservation of conditionally periodic motions at small change in Hamilton function."Reports of the USSR Academy of Sciences, 98(4), (1954): 527-530.

[2] Siegel C.L., "Vorlesungen uber Himmelsmechanik,"Berlin:Gottingen: Heidelberg: Springer-Verlag, 1956.

[3] Arnold V.I., "Small denominators. Proof of Theorem A.N. Kolmogorov on the conservation of conditionally-periodic motions with a small change in the Hamiltonian function,"SMS, 18(5), 1963:13-39.

[4] Arnold V.I., "Small denominators and the problem of stability in classical and celestial mechanics,"SMS, 18(6), (1963): 92-191.

[5] Moser J., "Convergent series expansion for quasi-periodic motions,"Mathem Ann, 169, (1967):136-176.

[6] Moser J., "A rapidly convergent iteration method and nonlinear partial differential equations,"Ann, Scuola Nozm Super, de Pisa, ser111 20(2), (1966): 65-315.

[7] Moser J., "Regularization of Kepler’s problem and averaging method on a manifold,"Communs Pure and Appl. Math. 23, (1970): 609-636.

[8] Jeffreys W., Moser J., "Quasi-periodic solutions three-body problem,"Actron.J., 7, (1966):568-578.

[9] Merman G.A., "New class of periodic solutions in Hill’s restricted problem,"Works of the Institute of Theoretical Astr. USSR Academy of Sciences 8, 5.

[10] Lika L.K., Ryabov Yu.A., "Construction of conditionally periodic solutions of canonical systems of differential equations,"Izvestiya Academy of Sciences of the Moldavian Soviet Socialist Republic. Series of physical and technical and mathematical sciences, 2, (1971):204-311.

[11] Grebennikov E.A., Ryabov Yu.A., "Resonances and small denominators in celestial mechanics,"M.Nauka, 1978. -pp. 126.

[12] Bogolyubov N.N. Mitropolsky Yu.A., Samoylenko A.M., "Accelerated Method of Convergence in Nonlinear Mechanics,"Kiev: "Naukova dumka 1969.

[13] Bari A., Brezis H., "Periodic solutions of nonlinear wave equations,"Comm. Pure Aple Math. 31(1), (1978):1-30.

[14] Feireist E., "On the existence of period o Solutions of a semilinear wave equation with a super linear forcing term,"Chechosl. Math. J., 38(1), (1988): 78-87.

[15] Pelyukh H.P., Syvak O.A., "Periodic Solutions of the Systems of Nonlinear Functional Equations,"Journal of Math. Sciences, 1, (2014): 92-95.

[16] Suleimenov "On the existence and stability of quasi-periodic solutions of a quasilinear system of differential equations,"Actual problems of mathematics and mathematical modeling are dedicated to the 50th anniversary of the creation of the Institute of Mathematics and Mechanics Academy of Sciences of the Kazakh SSR Almaty June 1-5, 2015.

[17] Suleimenov Zh., "The method of accelerated convergence for constructing conditional-periodical solutions,"Third International Conference on Analysis and Applied Mathematics ICAAM 2016.

[2] Siegel C.L., "Vorlesungen uber Himmelsmechanik,"Berlin:Gottingen: Heidelberg: Springer-Verlag, 1956.

[3] Arnold V.I., "Small denominators. Proof of Theorem A.N. Kolmogorov on the conservation of conditionally-periodic motions with a small change in the Hamiltonian function,"SMS, 18(5), 1963:13-39.

[4] Arnold V.I., "Small denominators and the problem of stability in classical and celestial mechanics,"SMS, 18(6), (1963): 92-191.

[5] Moser J., "Convergent series expansion for quasi-periodic motions,"Mathem Ann, 169, (1967):136-176.

[6] Moser J., "A rapidly convergent iteration method and nonlinear partial differential equations,"Ann, Scuola Nozm Super, de Pisa, ser111 20(2), (1966): 65-315.

[7] Moser J., "Regularization of Kepler’s problem and averaging method on a manifold,"Communs Pure and Appl. Math. 23, (1970): 609-636.

[8] Jeffreys W., Moser J., "Quasi-periodic solutions three-body problem,"Actron.J., 7, (1966):568-578.

[9] Merman G.A., "New class of periodic solutions in Hill’s restricted problem,"Works of the Institute of Theoretical Astr. USSR Academy of Sciences 8, 5.

[10] Lika L.K., Ryabov Yu.A., "Construction of conditionally periodic solutions of canonical systems of differential equations,"Izvestiya Academy of Sciences of the Moldavian Soviet Socialist Republic. Series of physical and technical and mathematical sciences, 2, (1971):204-311.

[11] Grebennikov E.A., Ryabov Yu.A., "Resonances and small denominators in celestial mechanics,"M.Nauka, 1978. -pp. 126.

[12] Bogolyubov N.N. Mitropolsky Yu.A., Samoylenko A.M., "Accelerated Method of Convergence in Nonlinear Mechanics,"Kiev: "Naukova dumka 1969.

[13] Bari A., Brezis H., "Periodic solutions of nonlinear wave equations,"Comm. Pure Aple Math. 31(1), (1978):1-30.

[14] Feireist E., "On the existence of period o Solutions of a semilinear wave equation with a super linear forcing term,"Chechosl. Math. J., 38(1), (1988): 78-87.

[15] Pelyukh H.P., Syvak O.A., "Periodic Solutions of the Systems of Nonlinear Functional Equations,"Journal of Math. Sciences, 1, (2014): 92-95.

[16] Suleimenov "On the existence and stability of quasi-periodic solutions of a quasilinear system of differential equations,"Actual problems of mathematics and mathematical modeling are dedicated to the 50th anniversary of the creation of the Institute of Mathematics and Mechanics Academy of Sciences of the Kazakh SSR Almaty June 1-5, 2015.

[17] Suleimenov Zh., "The method of accelerated convergence for constructing conditional-periodical solutions,"Third International Conference on Analysis and Applied Mathematics ICAAM 2016.

How to Cite

SULEIMENOV, Zh..
On the existence of a conditionally periodic solution of one quasilinear differential system in the critical case.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 100, n. 4, p. 8-17, jan. 2019. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/553>. Date accessed: 20 jan. 2021. doi: https://doi.org/10.26577/JMMCS-2018-4-553.
Section

Mathematics

Keywords
conditionally periodic, accelerated convergence, frequency, resonance