Geodesic
Lyapunov characteristics of oscillation, rotation, and wandering of solutions of differential systems
Trudy Seminara im. I.G. Petrovskogo, Trudy Seminara imeni I. G. Petrovskogo, Tome 31 (2016) no. 31, pp. 177-219 
I. N. Sergeev. Lyapunov characteristics of oscillation, rotation, and wandering of solutions of differential systems. Trudy Seminara im. I.G. Petrovskogo, Trudy Seminara imeni I. G. Petrovskogo, Tome 31 (2016) no. 31, pp. 177-219. http://geodesic.mathdoc.fr/item/TSP_2016_31_31_a8/
@article{TSP_2016_31_31_a8,
     author = {I. N. Sergeev},
     title = {Lyapunov characteristics of oscillation, rotation, and wandering of solutions of differential systems},
     journal = {Trudy Seminara im. I.G. Petrovskogo},
     pages = {177--219},
     year = {2016},
     volume = {31},
     number = {31},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TSP_2016_31_31_a8/}
}
TY  - JOUR
AU  - I. N. Sergeev
TI  - Lyapunov characteristics of oscillation, rotation, and wandering of solutions of differential systems
JO  - Trudy Seminara im. I.G. Petrovskogo
PY  - 2016
SP  - 177
EP  - 219
VL  - 31
IS  - 31
UR  - http://geodesic.mathdoc.fr/item/TSP_2016_31_31_a8/
LA  - ru
ID  - TSP_2016_31_31_a8
ER  - 
%0 Journal Article
%A I. N. Sergeev
%T Lyapunov characteristics of oscillation, rotation, and wandering of solutions of differential systems
%J Trudy Seminara im. I.G. Petrovskogo
%D 2016
%P 177-219
%V 31
%N 31
%U http://geodesic.mathdoc.fr/item/TSP_2016_31_31_a8/
%G ru
%F TSP_2016_31_31_a8

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A number of Lyapunov exponents are defined for solutions of linear systems on the half-line. These exponents are responsible for such properties of the solutions as oscillation, rotation, and wandering and are defined in terms of certain functionals applied to the solutions on finite intervals as a result of two operations: upper or lower averaging in time and minimization over all bases in the phase space. We consider important special cases of systems: those of a low order, autonomous systems, those associated with equations of an arbitrary order. We obtain a set of relations (equalities and inequalities) between the said exponents, together with their refined values in special cases. It is shown that this set is complete in the sense that it cannot be extended or strengthened by any other meaningful relation.

[1] Bylov B. F., Vinograd R. E., Grobman D. M., Nemytskii V. V., Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti, Nauka, M., 1966 | MR

[2] Sergeev I. N., “Kharakteristiki povorachivaemosti reshenii differentsialnykh sistem”, Differents. uravneniya, 50:10 (2014), 1353–1361 | DOI | Zbl

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

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

[5] Sergeev I. N., “Opredelenie kharakteristik vraschaemosti reshenii differentsialnykh sistem i uravnenii”, Differents. uravneniya, 49:11 (2013), 1501–1503

[6] Sergeev I. N., “Obobschennye kharakteristiki bluzhdaemosti reshenii differentsialnoi sistemy”, Differents. uravneniya, 49:11 (2013), 498–500

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

[8] Leontovich A. M., Melnikov M. S., “Ob ogranichennosti variatsii mnogoobraziya”, Tr. MMO, 14, 1965, 306–337

[9] Kuratovskii K., Topologiya, v. 1, Mir, M., 1966 | MR

[10] Hatanson I. P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR

[11] Sergeev I. N., “Koleblemost i bluzhdaemost reshenii differentsialnogo uravneniya vtorogo poryadka”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 2011 No 6, 21–26 | Zbl

[12] Tsoi S. V., “Primer nesovpadeniya polnoi i vektornoi chastot reshenii lineinoi sistemy”, Differents. uravneniya, 49:6 (2013), 815

[13] Kokushkin V. I., “Kharakteristiki koleblemosti i vraschaemosti reshenii lineinykh differentsialnykh sistem”, Differents. uravneniya, 50:10 (2014), 1406–1407 | DOI | MR | Zbl

[14] Burlakov D. S., Tsoi S. V., “Sovpadenie polnoi i vektornoi chastot reshenii lineinoi avtonomnoi sistemy”, Tr. seminara im. I. G. Petrovskogo, 30, 2014, 75–93

[15] Nemytskii V. V., “Kolebatelnye rezhimy mnogomernykh dinamicheskikh sistem”, Tr. Mezhdunarodnogo simpoziuma po nelineinym kolebaniyam, v. 2, Kiev, 1963, 308–314

[16] Bykov Ya. V., “Ob odnom klasse sistem obyknovennykh differentsialnykh uravnenii”, Differents. uravneniya, 1:11 (1965), 1449–1475

[17] Domshlak Yu. I., “O koleblemosti i nekoleblemosti reshenii vektornykh differentsialnykh uravnenii”, Differents. uravneniya, 7:6 (1971), 961–969 | MR

[18] Tonkov E. L., “Neostsillyatsiya i chislo pereklyuchenii v lineinoi sisteme, optimalnoi po bystrodeistviyu”, Differents. uravneniya, 9:12 (1973), 2180–2185 | Zbl

[19] Sergeev I. N., “Raspredelenie polnykh chastot i pokazatelei bluzhdaemosti v prostranstve reshenii lineinoi avtonomnoi sistemy”, Mezhdunarodnaya konferentsiya, posvyaschennaya 110-i godovschine so dnya rozhdeniya I. G. Petrovskogo, Tezisy dokladov, Izd-vo Mosk. un-ta, M.; OOO «INTUIT.RU», 2011, 342–343

[20] Sergeev I. N., “Opredelenie kharakteristicheskikh chastot lineinogo uravneniya”, Differents. uravneniya, 40:11 (2004), 1573 | MR | Zbl

[21] Sergeev I. N., “Opredelenie i svoistva kharakteristicheskikh chastot lineinogo uravneniya”, Tr. seminara im. I. G. Petrovskogo, 25, 2006, 249–294 | Zbl

[22] Sergeev I. N., “Opredelenie polnykh chastot reshenii lineinogo uravneniya”, Differents. uravneniya, 44:11 (2008), 1577

[23] Sergeev I. N., “Opredelenie polnykh chastot reshenii lineinoi sistemy”, Differents. uravneniya, 45:6 (2009), 908

[24] Sergeev I. N., “Opredelenie kharakteristik bluzhdaemosti reshenii lineinoi sistemy”, Differents. uravneniya, 46:6 (2010), 902