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## Archive of Issues

Russia Maikop
Year
2019
Volume
29
Issue
4
Pages
558-568
 Section Mathematics Title Properties of exponents of oscillation of linear autonomous differential system solutions Author(-s) Stash A.Kh.a Affiliations Adyghe State Universitya Abstract In this paper, we study various types of exponents of oscillation (upper or lower, strong or weak) of zeros, roots, hyperroots, strict and non-strict signs of non-zero solutions of linear homogeneous autonomous differential systems on the positive semi-axis. On the set of non-zero solutions of autonomous systems the relations between these exponents of oscillation are established. The spectra of the exponents of autonomous systems' oscillation are fully studied. It turned out that they directly depend on the roots of the corresponding characteristic polynomial of the system. As a consequence, spectra of all exponents of oscillation of autonomous systems with symmetric matrix are found. It is proved that they consist of a single zero value. In addition, a full description of the main values of the exponents of oscillation of such systems is given. These values for the exponents of oscillation of non-strict signs, roots and hyperroots coincided with the set of modules of imaginary parts of the system matrix's eigenvalues, and the exponents of oscillation of strict signs can consist of zero and the least, in absolute magnitude, imaginary part of the complex roots of the corresponding characteristic polynomial. Keywords differential equations, linear systems, oscillation, number of zeros, exponents of oscillation, Lyapunov exponents UDC 517.926 MSC 34C10 DOI 10.20537/vm190407 Received 20 August 2019 Language Russian Citation Stash A.Kh. Properties of exponents of oscillation of linear autonomous differential system solutions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 4, pp. 558-568. References Sergeev I.N. Definition and properties of characteristic frequencies of a linear equation, Journal of Mathematical Sciences, 2006, vol. 135, no. 1, pp. 2764-2793. https://doi.org/10.1007/s10958-006-0142-6 Sergeev I.N. The remarkable agreement between the oscillation and wandering characteristics of solutions of differential systems, Sbornik: Mathematics, 2013, vol. 204, no. 1, pp. 114-132. https://doi.org/10.1070/SM2013v204n01ABEH004293 Sergeev I.N. Oscillation and wandering characteristics of solutions of a linear differential systems, Izvestiya: Mathematics, 2012, vol. 76, no. 1, pp. 139-162. https://doi.org/10.1070/IM2012v076n01ABEH002578 Sergeev I.N. The complete set of relations between the oscillation, rotation and wandering indicators of solutions of differential systems, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2015, issue 2 (46), pp. 171-183 (in Russian). http://mi.mathnet.ru/eng/iimi318 Burlakov D.S., Tsoii S.V. Coincidence of complete and vector frequencies of solutions of a linear autonomous system, Journal of Mathematical Sciences, 2015, vol. 210, no. 2, pp. 155-167. https://doi.org/10.1007/s10958-015-2554-7 Stash A.Kh. Properties of full and vector frequencies of lax signs and roots of solutions of linear homogenous autonomous differential equations, Vestnik Adygeiskogo Gosudarstvennogo Universiteta. Seriya 4: Estestvenno-Matematicheskie i Tekhnicheskie Nauki, 2015, issue 3 (166), pp. 18-22 (in Russian). Stash A.Kh. Properties of complete and vector sign frequencies of solutions of linear autonomous differential equations, Differential Equations, 2014, vol. 50, no. 10, pp. 1418-1422. https://doi.org/10.1134/S0012266114100206 Tyrtyshnikov E.E. Matrichnyi analiz i lineinaya algebra (Matrix analysis and linear algebra), Moscow: Fizmatlit, 2007. Full text