Kazakh Mathematical Journal. 2019; : 55-70
Интегральное представление многопериодического решения линейной системы в одном критическом случае
Аннотация
Рассматривается задача существования и интегрального представления единственного многопериодического решения неоднородной линейной системы с постоянными коэффициентами и оператором дифференцирования D по направлению главной диагонали пространства временных переменных. Эта задача решена в некритическом случае, когда все собственные значения матрицы коэффициентов системы имеют отличные от нуля действительные части, причем метод исследования этого случая не был пригоден для изучения критического случая. Таким образом, проблема изучения критических случаев оставалась открытой. В данной работе доказывается существование решения задачи, когда матрица коэффициентов имеет несколько чисто мнимых собственных значений с простыми элементарными делителями, а свободный член системы обладает свойствами вещественной аналитичности по независимым переменным, изменяющимися в полосе действительной оси комплексной плоскости и периодичности с рационально несоизмеримыми частотами, причем частоты собственных колебаний и вынуждающей силы вместе удовлетворяют диофантовому условию сильной несоизмеримости. Установлено условие отсутствия ненулевого многопериодического решения однородной системы, соответствующей заданной системе. На этой основе построена матричная функция типа Грина, в терминах которой решается вопрос об интегральной структуре и существовании искомого единственного вещественно аналитического многопериодического решения. При исследовании задачи система линейной заменой расщепляется на подсистемы двух видов: а) несколько однотипных систем второго порядка критического характера и б) систему некритического случая. Поставленная задача решается для этих указанных подсистем в отдельности по описанной выше методике, а затем разработанный метод излагается в общей форме для исходной системы. В целом, в работе предложен новый метод исследования задачи существования и построения единственного многопериодического решения линейной системы уравнений с постоянными коэффициентами и одинаковым оператором дифференцирования D. Данный метод применим как в некритических, так и в критических случаях.
Список литературы
1. Lyapunov A.M. The general problem of stability of motion, GITTL, Moscow-Leningrad, 1950 (in Russian).
2. Malkin I.G. Methods of Lyapunov and Poincare in the theory of nonlinear oscillation, Editorial, Moscow, 2004 (in Russian).
3. Malkin I.G. Some problems of nonlinear oscillations theory, Editorial, Moscow, 2004 (in Russian).
4. Yakubovich V.A., Starzhinsky V.M. Linear differential equations with periodic coe ffi cients and their applications, Nauka, Moscow, 1972 (in Russian).
5. Demidovich B.P. Lectures on the mathematical theory of stability, Nauka, Moscow, 1967 (in Russian).
6. Vazov V. Asymptotic expansions of solutions of ordinary differential equations, Mir, Moscow, 1968 (in Russian).
7. Umbetzhanov D.U., Sartabanov Zh.A. On the necessary and su ffi cient condition for the multiperiodicity solutions of one system of partial differential equations with the same main part, Mathematics and Mechanics, 7:2 (1972), 22-27 (in Russian).
8. Khinchin A.Ja. Chain fractions, Nauka, Moscow, 1978 (in Russian).
9. Arnold V.I. Geometrical methods in the theory of ordinary differential equations, SpringerVerlag, 1983. https://doi.org10.1002/zamm.19860660910.
10. De la Llave R. A tutorial on KAM theory, Institute Computer research, Moscow-Izhevsk, 2003 (in Russian).
11. Dzhakalya G.E.O. Methods of perturbation theory for nonlinear systems, Institute Computer research, Nauka, Moscow, 1979 (in Russian).
12. Kharasakhal V.Kh. Almost-periodic solutions of ordinary differential equations, Nauka, Alma-Ata, 1970 (in Russian).
13. Umbetzhanov D.U. Almost multiperiodic solutions of partial differential equations, Nauka, Alma-Ata, 1979 (in Russian).
14. Sartabanov Zh. About one method of studying the periodic solutions of systems of partial differential equations of a special type, Bulletin of the Academy of Sciences of the Kazakh SSR. Ser. physical mat., 1 (1989), 42-49 (in Russian).
15. Mukhambetova, A.A., Sartabanov, Zh.A. Stability of solutions of the systems of differential equations with multidimensional time, Print A, Aktobe, 2007 (in Russian).
16. Kulzhumieva A.A., Sartabanov Zh.A. Periodic solutions of the systems of differential equations with multidimensional time, RIC WKSU, Uralsk, 2013 (in Russian).
17. Sartabanov, Z.A. The multi-period solution of a linear system of equations with the operator of differentiation along the main diagonal of the space of independent variables and delayed arguments, AIP Conference Proceedings, 1880 (2017), 040020. https://doi.org/10.1063/1.5000636.
18. Kolmogorov A.N. On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR, 98:4 (1954), 527-530 (in Russian).
19. Arnold V. I. Small denominators. I. Mapping the circle onto itself, Izv. Akad. Nauk SSSR Ser. Mat., 25:1 (1961), 21-86 (in Russian).
20. Arnold V. I. Proof of a theorem of A.?N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russian Math. Surveys, 18:5 (1963), 13-40 (in Russian).
21. Arnold V. I. Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Surveys, 18:6 (1963), 91-192 (in Russian).
22. Moser J. Lectures on Hamiltonian systems, Mir, Moscow, 1973 (in Russian).
23. Moser J. KAM theory and problems of stability, SIC “Regular and chaotic. dynamics”, Izhevsk, 2001 (in Russian).
24. Bogoliubov N.N., Mitropolsky Yu.A., Samoilenko A.M. Method of Accelerated Convergence in Nonlinear Mechanics, Springer Verlag, Delhi and Berlin, 1976.
25. Sartabanov Z.A., Omarova, B.Z. Multiperiodic solutions of autonomous systems with operator of differentiation on the Lyapunov’s vector field, AIP Conference Proceedings, 1997 (2018), 020041. https://doi.org/10.1063/1.5049035.
26. Sartabanov Zh.A., Omarova B.Zh. Multiperiodic solutions of the one autonomous system of equations with the operator of differentiation with respect to spatial and time variables, Scientific journal Vestnik of Aktobe’s K.Zhubanov Regional State University, 51:1 (2018), 60-64.
27. Sartabanov Zh.A., Omarova B.Zh. On multi-periodic solutions of quasilinear autonomous systems with operator of differentiation on the Lyapunov’s vector field, Bulletin of the Karaganda University-Mathematics, 94:2 (2019), 70-83. https://doi.org/10.31489/2019M2/70-83.
28. Sartabanov Zh.A., Omarova B.Zh., Kerimbekov A. Research of multiperiodic solutions of perturbed linear autonomous systems with differentiation operator on the vector field, News of the National Academy of Sciences of the Republic of Kazakhstan.Physico-Mathematical Series, 328:6 (2019), 63-79. https://doi.org/10.32014/2019.2518-1726.74.
Kazakh Mathematical Journal. 2019; : 55-70
Integral representation of multiperiodic solutions of a linear system in one critical case
Abstract
The problem of the existence and integral representation of a unique multiperiodic solution of an inhomogeneous linear system with constant coefficients and a differentiation operator D on the direction of the main diagonal of the space of time variables were considered. This problem was solved in non-critical case when all eigenvalues of the matrix of coefficients of the system have non-zero real parts; moreover, the method of studying this case was not suitable for studying a critical case. Thus, critical cases remained open. This proves the existence of a solution to the problem when the matrix of coefficients has several pure imaginary eigenvalues with simple elementary divisors, and a free member of the system has the properties of real analyticity in independent variables that change in the strip of the real axis of the complex plane, and periodicity with rationally incommensurable frequencies. Moreover, the frequencies of the eigenvalues oscillations and excitation forces together satisfy the Diophantine condition of strong incommensurability. The condition for the absence of a nonzero multiperiodic solution of the homogeneous system corresponding to the given system is established. On this basis, the Green-type matrix function is constructed, in terms of which the question of the integral structure and existence of the required unique real analytic multiperiodic solution is solved. When studying the problem of a linear replacement system, it splits into two types of subsystems: a) several similar systems of the second order of critical nature and b) a system of non-critical cases. The problem is solved for these indicated subsystems individually according to the method described above, and then the developed method is described in general form for the original system. In general, the work proposes a new method for studying the problem of the existence and construction of a unique multiperiodic solution of the linear system of equations with constant coefficients and the same differentiation operator D. The method which is applicable in both non-critical and critical cases.
References
1. Lyapunov A.M. The general problem of stability of motion, GITTL, Moscow-Leningrad, 1950 (in Russian).
2. Malkin I.G. Methods of Lyapunov and Poincare in the theory of nonlinear oscillation, Editorial, Moscow, 2004 (in Russian).
3. Malkin I.G. Some problems of nonlinear oscillations theory, Editorial, Moscow, 2004 (in Russian).
4. Yakubovich V.A., Starzhinsky V.M. Linear differential equations with periodic coe ffi cients and their applications, Nauka, Moscow, 1972 (in Russian).
5. Demidovich B.P. Lectures on the mathematical theory of stability, Nauka, Moscow, 1967 (in Russian).
6. Vazov V. Asymptotic expansions of solutions of ordinary differential equations, Mir, Moscow, 1968 (in Russian).
7. Umbetzhanov D.U., Sartabanov Zh.A. On the necessary and su ffi cient condition for the multiperiodicity solutions of one system of partial differential equations with the same main part, Mathematics and Mechanics, 7:2 (1972), 22-27 (in Russian).
8. Khinchin A.Ja. Chain fractions, Nauka, Moscow, 1978 (in Russian).
9. Arnold V.I. Geometrical methods in the theory of ordinary differential equations, SpringerVerlag, 1983. https://doi.org10.1002/zamm.19860660910.
10. De la Llave R. A tutorial on KAM theory, Institute Computer research, Moscow-Izhevsk, 2003 (in Russian).
11. Dzhakalya G.E.O. Methods of perturbation theory for nonlinear systems, Institute Computer research, Nauka, Moscow, 1979 (in Russian).
12. Kharasakhal V.Kh. Almost-periodic solutions of ordinary differential equations, Nauka, Alma-Ata, 1970 (in Russian).
13. Umbetzhanov D.U. Almost multiperiodic solutions of partial differential equations, Nauka, Alma-Ata, 1979 (in Russian).
14. Sartabanov Zh. About one method of studying the periodic solutions of systems of partial differential equations of a special type, Bulletin of the Academy of Sciences of the Kazakh SSR. Ser. physical mat., 1 (1989), 42-49 (in Russian).
15. Mukhambetova, A.A., Sartabanov, Zh.A. Stability of solutions of the systems of differential equations with multidimensional time, Print A, Aktobe, 2007 (in Russian).
16. Kulzhumieva A.A., Sartabanov Zh.A. Periodic solutions of the systems of differential equations with multidimensional time, RIC WKSU, Uralsk, 2013 (in Russian).
17. Sartabanov, Z.A. The multi-period solution of a linear system of equations with the operator of differentiation along the main diagonal of the space of independent variables and delayed arguments, AIP Conference Proceedings, 1880 (2017), 040020. https://doi.org/10.1063/1.5000636.
18. Kolmogorov A.N. On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR, 98:4 (1954), 527-530 (in Russian).
19. Arnold V. I. Small denominators. I. Mapping the circle onto itself, Izv. Akad. Nauk SSSR Ser. Mat., 25:1 (1961), 21-86 (in Russian).
20. Arnold V. I. Proof of a theorem of A.?N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russian Math. Surveys, 18:5 (1963), 13-40 (in Russian).
21. Arnold V. I. Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Surveys, 18:6 (1963), 91-192 (in Russian).
22. Moser J. Lectures on Hamiltonian systems, Mir, Moscow, 1973 (in Russian).
23. Moser J. KAM theory and problems of stability, SIC “Regular and chaotic. dynamics”, Izhevsk, 2001 (in Russian).
24. Bogoliubov N.N., Mitropolsky Yu.A., Samoilenko A.M. Method of Accelerated Convergence in Nonlinear Mechanics, Springer Verlag, Delhi and Berlin, 1976.
25. Sartabanov Z.A., Omarova, B.Z. Multiperiodic solutions of autonomous systems with operator of differentiation on the Lyapunov’s vector field, AIP Conference Proceedings, 1997 (2018), 020041. https://doi.org/10.1063/1.5049035.
26. Sartabanov Zh.A., Omarova B.Zh. Multiperiodic solutions of the one autonomous system of equations with the operator of differentiation with respect to spatial and time variables, Scientific journal Vestnik of Aktobe’s K.Zhubanov Regional State University, 51:1 (2018), 60-64.
27. Sartabanov Zh.A., Omarova B.Zh. On multi-periodic solutions of quasilinear autonomous systems with operator of differentiation on the Lyapunov’s vector field, Bulletin of the Karaganda University-Mathematics, 94:2 (2019), 70-83. https://doi.org/10.31489/2019M2/70-83.
28. Sartabanov Zh.A., Omarova B.Zh., Kerimbekov A. Research of multiperiodic solutions of perturbed linear autonomous systems with differentiation operator on the vector field, News of the National Academy of Sciences of the Republic of Kazakhstan.Physico-Mathematical Series, 328:6 (2019), 63-79. https://doi.org/10.32014/2019.2518-1726.74.
