A.Kh. Stash. On essential values of exponents of oscillation for solutions of a linear homogeneous two-dimensional differential system ... P. 157-171

In this paper, we study various types of exponents of oscillations for solutions of linear homogeneous differential systems with continuous bounded coefficients. The calculation of the exponents of oscillation is carried out by averaging the number of zeros (or signs, or roots, or hyper roots) of the projection of a solution~$x$ of a differential system onto any straight line, and this line is chosen so that the resulting average value is minimal: if the minimization is performed before (after) the averaging, then weak (strong, respectively) exponents of oscillation are obtained. In the calculation of the exponents of oscillation for a solution $y$ of a linear homogeneous $n$-th order differential equation, a transition to the vector function $x=(y, \dot y,\dots, y^{(n-1)})$ is carried out. In the first part of the paper, for any preassigned natural number $N$, a two-dimensional periodic linear differential system is constructed, which has the property that its spectra of all upper and lower strong and weak exponents of oscillation of strict and nonstrict signs, zeros, roots, and hyper roots contain the same set, consisting of $N$ different essential values, both metrically and topologically. Moreover, all these values are implemented on the same set of solutions of the constructed system, that is, for each solution from this set, all the exponents of oscillations coincide with each other. In the second part of the paper, a similar theorem on the existence of a two-dimensional differential system with a countable set of essential (both metrically and topologically) values of exponents of oscillation is proved. In constructing the mentioned systems and proving the required results, we use analytical methods of the qualitative theory of differential equations and methods of the theory of perturbations of solutions of linear differential systems, in particular, the author's technique for controlling the fundamental matrix of solutions of such systems in one special case.

Keywords: differential equation, linear system, oscillation, number of zeros, exponents of oscillation, Sergeev's frequency

Received February 2, 2023

Revised April 17, 2023

Accepted April 24, 2023

Aydamir Khazretovich Stash, Cand. Sci. (Phys.-Math.), Dean of the Faculty of Mathematics and Computer Science, Adyghe state University, Maykop, 385000 Russia, e-mail: aidamir.stash@gmail.com

REFERENCES

1.   Sergeev I.N. Definition and properties of characteristic frequencies of a linear equation. J. Math. Sci., 2006, vol. 135, no. 1, pp. 2764–2793. doi: 10.1007/s10958-006-0142-6

2.   Sergeev I.N. Oscillation and wandering characteristics of solutions of a linear differential systems. Izvestiya: Mathematics, 2012, vol. 76, no. 1, pp. 139–162. doi: 10.1070/IM2012v076n01ABEH002578

3.   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. doi: 10.1070/SM2013v204n01ABEH004293

4.   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, iss. 2 (46), pp. 171–183 (in Russian).

5.   Sergeev I.N. Lyapunov characteristics of oscillation, rotation, and wandering of solutions of differential systems. J. Math. Sci., 2018, vol. 234, no. 4, pp. 497–522. doi: 10.1007/s10958-018-4025-4

6.   Sergeev I.N.Oscillation, rotation, and wandering exponents of solutions of differential systems. Math. Notes, 2016, vol. 99, no. 5, pp. 729–746. doi: 10.1134/S0001434616050114

7.   Barabanov E.A., Voidelevich A.S. Remark on the theory of Sergeev frequencies of zeros, signs and roots for solution of linear differential equation: I. Diff. eq., 2016, vol. 52, no. 10, pp. 1249–1267. doi: 10.1134/S0012266116100013

8.   Barabanov E.A., Voidelevich A.S. Remark on the theory of Sergeev frequencies of zeros, signs and roots for solution of linear differential equation:II, Diff. eq., 2016, vol. 52. no.  12, pp. 1523–1538. doi:10.1134/S0012266116120016

9.   Bykov V.V. On the Baire classification of Sergeev frequencies of zeros and roots of solution of linear differential equation, Diff. eq., 2016, vol. 52. no. 4, pp. 413–420. doi: 10.1134/S0012266116040029

10.   Barabanov E.A., Voidelevich A.S. Spectra of the upper Sergeev frequencies of zeros and signs of linear differential equation, Doklady NAN Belarusi, 2016, vol. 60, no. 1, pp. 24–31 (in Russian).

11.   Voidelevich A.S. On spectra of upper Sergeev frequencies of linear differential equation, Zhurnal Belarusskogo Gosudarstvennogo Instituta. Matematika. Informatika, 2019, no. 1, pp. 28–32 (in Russian).

12.   Sergeev I.N. Metrically typical and essential values of exponents of linear systems, Differencialnie uravneniya. 2011, vol. 47, no. 11, pp. 1661–1662 (in Russian).

13.   Sergeev I.N. Topologically typical and essential values of exponents of linear systems, Differencialnie uravneniya. 2012, vol. 48, no. 11. pp. 1567–1568 (in Russian).

14.   Stash A.Kh. The absence of residual property for strong exponents of oscillation of linear systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 2021, vol. 31, iss. 1, pp. 59–69 (in Russian).

15.   Stash A. Kh. Existence of a two-dimensional linear system with continual spectra of total and vector frequencies. Diff. eq., 2015, vol. 51, no. 1, pp. 146–148. doi:10.1134/S0374064115010161

16.   Burlakov D.S., Tsoii S.V. Coincidence of complete and vector frequencies of solutions of a linear autonomous system. J. Math. Sci., 2015, vol. 210, no. 2, pp. 155–167. doi: 10.1007/s10958-015-2554-7

17.   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, iss. 4, pp. 558–568 (in Russian). doi: 10.20537/vm190407

18.   Stash A.Kh. On finite spectra of full and vector frequencies of linear two-dimensional differential periodic system. Vestnik Adygeiskogo gosudarstvennogo universiteta. Seriya 4: Estestvenno–matematicheskie i tekhnicheskie nauki, 2014, iss. 1 (133), pp. 30–36 (in Russian).

19.   Stash A.Kh. About calculating ranges of full and vector frequencies of the linear two-dimensional differential system. Vestnik Adygeiskogo gosudarstvennogo universiteta. Seriya 4: Estestvenno–matematicheskie i tekhnicheskie nauki, 2014, iss. 2 (137), pp. 23–32 (in Russian).

20.   Filippov A.F. Vvedenie v teoriju differencial’nyh uravnenij [Introduction to the theory of differential equations]. Moscow: Editorial URSS, 2004. 240 p.

21.   Shishlyannikov E.M. Two dimensional differential systems with arbitrary finite spectra of wandering exponent. Moscow Univ. Math. Bull., 2017, vol 72, no. 5, pp. 192–198. doi: 10.3103/S0027132217050023

Cite this article as: A.Kh. Stash. On essential values of exponents of oscillation for solutions of a linear homogeneous two-dimensional differential system. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 2, pp. 157–171; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 321, Suppl. 1, pp. S216–S229.