Relationship between components of a strongly irregular quasiperiodic solutions of the linear homogeneous algebraic system
Abstract
We study a linear homogeneous algebraic system with the quasiperiodic matrix of coefficients for the existence strongly irregular quasiperiodic solution. If there is such a solution, then there is a linear relationship between its components. An algorithm for finding this dependence is given.
Keywords
linear algebraic system,
quasiperiodic coefficients matrix,
quasiperiodic solutions,
strong irregular,
components dependence
Supporting agencies: The work was performed at the Institute of Mathematics of the National Academy of Sciences of Belarus on assignment 1.2.05 "Nonstationary differential systems with unstable behavior of reductions" GPNI "Convergence–2025" (subprogram "Mathematical models and methods") within the framework of a grant from the President of the Republic Belarus
About the Author
A. K. Demenchuk
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Minsk
References
1. Samoilenko A. S. Quasiperiodic solutions of linear systems algebraic equations with quasiperiodic coefficients Analytical methods for studying solutions of nonlinear differential equations. Kiev, Inst. of Math. of Acad. of Sc. of Ukrainian SSR, 1975, pp. 5–26 (in Russian).
2. Malkin I. G. Some problems of the theory of nonlinear oscillations. Moscow, GITTL, 1956 (in Russian).
3. Hajassi T. Forced oscillations in nonlinear systems. Moscow, IIL, 1957 (in Russian).
4. Erugin N. P. Linear systems of ordinary differential equations with periodic and quasiperiodic coefficients. Minsk, Izd. Acad. of Sc. of BSSR, 1963 (in Russian).
5. Hale J. Oscillations in nonlinear systems. Moscow, Mir, 1966 (in Russian).
6. Kasnoselskii М. А., Burd V. Sh., Kolesovсов Yu. S. Nonlinear almost periodic oscillations. Moscow, Nauka, 1970 (in Russian).
7. Yakubovich V. A., Starginskii V. M. Linear differential equations with periodic coefficients and their applications. Moscow, Nauka, 1972 (in Russian).
8. Zubov V. I. Theory of oscillations. Moscow, Vysshaja Shkola, 1979 (in Russian).
9. Papaleksi N. D. On one case of parametrically related systems. Journ. Of Phys. Acad. of Sc. of USSR, 1939, vol. 1, pp. 373–379 (in Russian).
10. Penner D. I., Duboshinskii Ya. B., Duboshinskii D. B. Oscillations with self-regulating interaction time. Doklady Acad. of Sc. of USSR, 1972, vol. 204, no. 5, pp. 1065–1066.
11. Landa P. S., Duboshinskii Ya. B. Self-oscillatory systems with high-frequency energy sources. Uspekhi Fiz. Nauk, 1989, vol. 158, no. 4, pp. 729–742 (in Russian).
12. Massera J. L. Observaciones sobre les soluciones periodicas de ecuaciones diferenciales. Bol. de la Facultad de Ingenieria, 1950, vol. 4, no. 1, pp. 37–45.
13. Kurzweil Ya., Veyvoda O. On periodic and almost periodic solutions of systems of ordinary differential equations. Czechosl. Math. J., 1955, vol. 5, no. 3, pp. 362–370.
14. Erugin N. P. On periodic solutions of differential equations. Priklad. Math. and Mech., 1956, vol. 20, no. 1, pp. 148–152 (in Russian).
15. Erugin N. P. On periodic solutions of linear homogeneous systems of differential equations. Doklady Acad. of Sc. of BSSR, 1962, vol. 6, no. 7, pp. 407–410 (in Russian).
16. Gaishun I. V. Total derivative equations with periodic coefficients. Doklady Acad. of Sc. of BSSR, 1979, vol. 23, no. 8, pp. 684–686 (in Russian).
17. Grudo E. I. On periodic solutions with incommensurable periods of periodic differential systems. Differ. Uravn., 1986, vol. 22, no. 9, pp. 1499–1504 (in Russian).
18. Grudo E. I. Periodic systems having solutions with incommensurate periods in relation to the period of the system. Uspekhi Math. Nauk, 1986, vol. 41, no. 4, p. 200 (in Russian).
19. Grudo E. I., Demenchuk A. K. On periodic solutions with incommensurate periods of linear inhomogeneous periodic differential systems. Differ. Uravn., 1987, vol. 23, no. 3, pp. 409–416 (in Russian).
20. Borukhov V. T. Strongly invariant subspaces of non-autonomous linear periodic systems and solutions whose period is incommensurable with the period of the system itself. Differential Equations, 2018, vol. 54, pp. 578–585. doi: 10.1134/S0012266118050026
21. Demenchuk А. К. Partially irregular almost periodic solutions of ordinary differential systems. Math. Bohemica, 2001, vol. 126, no. 1, pp. 221–228.
22. Bohr G. Almost periodic functions. Moscow–Leningrad, GTTIH, 1934 (in Russian).
23. Demenchuk A. K. Asynchronous oscillations in differential systems. Conditions of existence and control. Saarbrucken, Lambert Academic Publishing, 2012 (in Russian).
24. Horn R., Johnson Ch. Matrix analysis. Moscow, Nauka, 1989 (in Russian).
Review
For citations:
Demenchuk A.K. Relationship between components of a strongly irregular quasiperiodic solutions of the linear homogeneous algebraic system. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2024;32(1):64-73. (In Russ.)
Views: 38
Email this article 