Geodesic
On the continuum spectra of the exponents of linear homogeneous differential systems
Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 141, pp. 60-67 Cet article a éte moissonné depuis la source Math-Net.Ru

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The research subject of this work is at the junction of two sections of the qualitative theory of differential equations, namely: the theory of Lyapunov exponents and the theory of oscillation. In this paper, we study the spectra (i. e., sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential systems with coefficients continuous on the positive semiaxis. For any $n\ge 2$, the existence of an $n$-dimensional differential system with continuum spectra of the oscillation exponents is established. For even $n$, the spectra of all the oscillation exponents fill the same segment of the numerical axis with predetermined arbitrary positive incommensurable ends, and for odd $n$, zero is added to the indicated spectra. It turns out that for each solution of the constructed differential system, all the oscillation exponents coincide with each other. When proving the results of this work, the cases of even and odd $n$ are considered separately. The results obtained are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential systems.
Keywords: differential equations, linear homogeneous differential systems, spectrum of the system exponent, number of zeros of solution, exponents of oscillation, Sergeev's frequencies.
Mots-clés : oscillation of solution 
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A. Kh. Stash. On the continuum spectra of the exponents of linear homogeneous differential systems. Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 141, pp. 60-67. http://geodesic.mathdoc.fr/item/VTAMU_2023_28_141_a5/

[1] I. N. Sergeev, “Opredelenie i svoistva kharakteristicheskikh chastot lineinogo uravneniya”, Trudy seminara imeni I. G. Petrovskogo, 25, Izd-vo Mosk. un-ta, M., 2006, 249–294

[2] I. N. Sergeev, “Zamechatelnoe sovpadenie kharakteristik koleblemosti i bluzhdaemosti reshenii differentsialnykh sistem”, Matematicheskii sbornik, 204:1 (2013), 119–138 | DOI | MR

[3] I. N. Sergeev, “Kharakteristiki koleblemosti i bluzhdaemosti reshenii lineinoi differentsialnoi sistemy”, Izvestiya RAN. Seriya matematicheskaya, 76:1 (2012), 149–172 | DOI | MR

[4] I. N. Sergeev, “The complete set of relations between the oscillation, rotation and wandering indicators of solutions of differential systems”, Proceedings of the Institute of Mathematics and Computer Science of UdSU, 2015, no. 2(46), 171–183 (In Russian)

[5] I. N. Sergeev, “Pokazateli koleblemosti, vraschaemosti i bluzhdaemosti reshenii differentsialnykh sistem”, Matematicheskie zametki, 99:5 (2016), 732–751 | DOI | MR

[6] I. N. Sergeev, “Lyapunovskie kharakteristiki koleblemosti, vraschaemosti i bluzhdaemosti reshenii differentsialnykh sistem”, Trudy Seminara im. I. G. Petrovskogo, 31 (2016), 177–219, Izd-vo Mosk. un-ta, M. ; I. N. Sergeev, “Lyapunov characteristics of oscillation, rotation, and wandering of solutions of differential systems”, Journal of Mathematical Sciences, 234:4 (2018), 497–522 | DOI | MR

[7] I. N. Sergeev, “Koleblemost, vraschaemost i bluzhdaemost reshenii lineinykh differentsialnykh sistem”, Trudy Mezhdunarodnogo simpoziuma «Differentsialnye uravneniya—2016», Perm, 17–18 maya 2016, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 132, VINITI RAN, M., 2017, 117–121 ; I. N. Sergeev, “Oscillation, rotation, and wandering of solutions to linear differential systems”, Journal of Mathematical Sciences, 230:5 (2018), 770–774 | DOI | MR

[8] I. N. Sergeev, “O pokazatelyakh koleblemosti, vraschaemosti i bluzhdaemosti differentsialnykh sistem, zadayuschikh povoroty ploskosti”, Vestnik MGU imeni M. V. Lomonosova. Seriya 1: Matematika. Mekhanika, 2019, no. 1, 21–26

[9] E. A. Barabanov, A. S. Voidelevich, “K teorii chastot Sergeeva nulei, znakov i kornei reshenii lineinykh differentsialnykh uravnenii. I”, Differentsialnye uravneniya, 52:10 (2016), 1302–1320 | DOI

[10] E. A. Barabanov, A. S. Voidelevich, “K teorii chastot Sergeeva nulei, znakov i kornei reshenii lineinykh differentsialnykh uravnenii. II”, Differentsialnye uravneniya, 52:12 (2016), 1595–1609 | DOI

[11] V. V. Bykov, “O berovskoi klassifikatsii chastot Sergeeva nulei i kornei reshenii lineinykh differentsialnykh uravnenii”, Differentsialnye uravneniya, 52:4 (2016), 419–425 | DOI | MR

[12] A. S. Voidelevich, “On spectra of upper Sergeev frequencies of linear differential equation”, Journal of the Belarusian State University. Mathematics. Computer science, 2019, no. 1, 28–32 (In Russian) | MR

[13] A. Kh. Stash, “On finite spectra of full and vector frequencies of linear two-dimensional differential periodic system”, Bulletin of the Adyghe State University. Series. Natural-mathematical and technical sciences, 2014, no. 1(133), 30–36 (In Russian)

[14] A. Kh. Stash, “About calculating ranges of full and vector frequencies of the linear two-dimensional differential system”, Bulletin of the Adyghe State University. Series. Natural-mathematical and technical sciences, 2014, no. 2(137), 23–32 (In Russian)

[15] D. S. Burlakov, S. V. Tsoi, “Sovpadenie polnoi i vektornoi chastot reshenii lineinoi avtonomnoi sistemy”, Trudy seminara im. I. G. Petrovskogo, no. 30, 2014, 75–93 ; D. S. Burlakov, S. V. Tsoii, “Coincidence of complete and vector frequencies of solutions of a linear autonomous system”, Journal of Mathematical Sciences, 210, no. 2 (2015), 155–167 | DOI | MR

[16] A. Kh. Stash, “Properties of exponents of oscillation of linear autonomous differential system solutions”, Bulletin of the Udmurt University. Mathematics. Mechanics. Computer Science, 29:4 (2019), 558–568 (In Russian) | MR

[17] A. Kh. Stash, “Suschestvovanie dvumernoi lineinoi sistemy s kontinualnymi spektrami polnykh i vektornykh chastot”, Differentsialnye uravneniya, 51:1 (2015), 143–144 | DOI | MR

[18] A. Yu. Goritskii, T. N. Fisenko, “Kharakteristicheskie chastoty nulei summy dvukh garmonicheskikh kolebanii”, Differentsialnye uravneniya, 48:4 (2012), 479–486 | MR

[19] M. V. Smolentsev, “Primer periodicheskogo differentsialnogo uravneniya tretego poryadka, spektr chastot kotorogo soderzhit otrezok”, Differentsialnye uravneniya, 50:10 (2014), 1413–1417 | DOI | MR