Abstract
The system of the equations with differential operator in the directions of a multiperiodic potential vector field are considered. The condition is given that determines the non-vorticity of the vector field. It is assumed that the coordinates of the vector field have the properties of periodicity and smoothness, where the periods are rationally incommensurable positive constants. The conditionally periodic structure of the characteristics is studied. The characteristics of the given differential operator are constructed, which leads to the solution of the functional equation. The case of \(m=2\) is investigated in more detail. The results are illustrated on the specific example. Then linear and nonlinear oscillations are investigated on the vortex-free multiperiodic vector field. It is assumed that the matrix and the given vector-function satisfy the conditions of periodicity and smoothness. It is established that the linear and quasilinear systems have a unique multiperiodic solution. In conclusion, quasiperiodic oscillations generated by multiperiodic oscillations on the vortex-free vector field are investigated.
REFERENCES
V. H. Kharasahal, Almost Periodic Solutions of the Ordinary Differential Equations (Nauka, Alma-Ata, 1970) [in Russian].
D. U. Umbetzhanov, Almost Multiperiodic Solutions of the Partial Differential Equations (Nauka, Alma-Ata, 1979) [in Russian].
G. V. Demidenko, ‘‘On the existence of periodic solutions to one class of systems of nonlinear differential equations,’’ Lobachevskii J. Math. 42, 3336–3343 (2021).
G. V. Demidenko, ‘‘On conditions for exponential dichotomy of systems of linear differential equations with periodic coefficients,’’ Int. J. Dyn. Syst. Differ. Equat. 6, 63–74 (2016).
N. T. Orumbayeva, A. T. Assanova, and A. B. Keldibekova, ‘‘On an algorithm of finding an approximate solution of a periodic problem for a third-order differential equation,’’ Euras. Math. J. 13, 69–85 (2022).
A. T. Assanova and S. S. Kabdrakhova, ‘‘Modification of the Euler polygonal method for solving a semi-periodic boundary value problem for pseudo-parabolic equation of special type,’’ Mediterr. J. Math. 17 (41), 109 (2020).
A. T. Assanova, Z. K. Dzhobulaeva, and A. E. Imanchiyev ‘‘A multi-point initial problem for a non-classical system of a partial differential equations,’’ Lobachevskii J. Math. 41, 1031–1042 (2020).
A. I. Aristov, ‘‘Exact solutions of three nonclassical equations, and their construction with Maple system,’’ Lobachevskii J. Math. 40, 851–860 (2019).
M. V. Falaleev and E. Y. Grazhdantseva, ‘‘Generalized solutions of differential equations with the derivatives of functionals in Banach spaces,’’ Lobachevskii J. Math. 42, 3626–3636 (2021).
T. K. Yuldashev, ‘‘On a boundary-value problem for a fourth-order partial integro-differential equation with degenerate kernel,’’ J. Math. Sci. 245, 508–523 (2020).
T. K. Yuldashev, ‘‘Nonlinear optimal control of thermal processes in a nonlinear inverse problem,’’ Lobachevskii J. Math. 41, 124–136 (2020).
T. K. Yuldashev and O. Kh. Abdullaev, ‘‘Unique solvability of a boundary value problem for a loaded fractional parabolic-hyperbolic equation with nonlinear terms,’’ Lobachevskii J. Math. 42, 1113–1123 (2021).
T. K. Yuldashev, B. I. Islomov, and A. A. Abdullaev, ‘‘On solvability of a Poincare–Tricomi type problem for an elliptic-hyperbolic equation of the second kind,’’ Lobachevskii J. Math. 42, 663–675 (2021).
T. K. Yuldashev, B. I. Islomov, and E. K. Alikulov, ‘‘Boundary-value problems for loaded third-order parabolic-hyperbolic equations in infinite three-dimensional domains,’’ Lobachevskii J. Math. 41, 926–944 (2020).
T. K. Yuldashev, B. J. Kadirkulov, and R. A. Bandaliyev, ‘‘On a mixed problem for Hilfer type fractional differential equation with degeneration,’’ Lobachevskii J. Math. 43, 263–274 (2022).
N. N. Bogolyubov, Yu. A. Mitropolskii, and A. M. Samoilenko, The Method of Accelerated Convergence in Nonlinear Mechanics (Naukova Dumka, Kiev, 1969) [in Russian].
Yu. A. Mitropolskii, A. M. Samoilenko, and D. I. Martynyuk, Systems of Evolution Equations with Periodic and Conditionally Periodic Coefficients (Naukova Dumka, Kiev, 1984) [in Russian].
A. M. Samoilenko, Elements of the Mathematical Theory of Multifrequency Oscillations. Invariant Tors (Nauka, Moscow, 1987) [in Russian].
J. Moser, Lectures on Hamiltonian Systems, and Rigorous and Formal Stability of Orbits About an Oblate Planet (Am. Math. Soc., Providence, RI, 1989).
Zh. A. Sartabanov, ‘‘Conditions of periodicity of the solutions of the differential systems with multivariate time,’’ Izv. NAN RK, Ser. Fiz.-Mat. 3, 44–48 (2004).
A. A. Kulzhumiyeva and Zh. A. Sartabanov, ‘‘On multiperiodic integrals of a linear system with the differentiation operator in the direction of the main diagonal in the space of independent variables,’’ Eur. Math. J. 8, 67–75 (2017).
A. A. Kulzhumiyeva and Zh. A. Sartabanov, Periodic Solutions of System of Differential Equations with Multivariate Time (Center of M. Utemissov WKSU, Uralsk, 2020) [in Russian].
A. A. Kulzhumiyeva and Zh. A. Sartabanov, ‘‘General bounded multiperiodic solutions of linear equation with differential operator in the direction of the main diagonal,’’ Bull. Karag. Univ. — Math. 92 (4), 44–53 (2018).
A. A. Kulzhumiyeva and Zh. A. Sartabanov, ‘‘On reducibility of linear \(D_{e}\)-system with constant coefficients on the diagonal to \(D_{e}\)-system with Jordan matrix in the case of equivalence of its higher order one equation,’’ Vestn. Karag. Univ. — Mat. 84 (4), 88–93 (2016).
A. A. Kulzhumiyeva and Zh. A. Sartabanov, ‘‘Reduction of linear homogeneous \(D_{e}\)-systems to the Jordan canonical form,’’ Izv. NAN RK, Ser. Fiz.-Mat. 315 (5), 5–12 (2017).
A. A. Kulzhumiyeva and Zh. A. Sartabanov, ‘‘Integration of a linear equation with differential operator, corresponding to the main diagonal in the space of independent variables, and coefficients, constant on the diagonal,’’ Russ. Math. 63 (6), 29–41 (2019).
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by A. T. Assanova)
Rights and permissions
About this article
Cite this article
Kulzhumiyeva, A.A., Sartabanov, Z. Multiperiodic Solutions of Systems of the Equations with Differential Operator in the Direction of a Vector Field. Lobachevskii J Math 43, 3205–3215 (2022). https://doi.org/10.1134/S1995080222140207
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080222140207