O.N. Masina1, O.V. Druzhinina2, E.V. Lisovsky3

1, 2 Bunin Yelets State University (Yelets, Russia)
2 FRC «Computer Science and Control» of RAS (Moscow)
2 V.A. Trapeznikov Institute of Control Sciences of RAS (Moscow)
3 Kaluga branch of the Moscow State Technical University n. a. N.E. Bauman
1olga121@inbox.ru; 2 ovdruzh@mail.ru; 3levgenijv@gmail.com

The development of methods of stability analysis and the study of various types of dynamic systems stability belongs to an urgent scientific direction. Among the problems solved in the framework of this direction, an important place is occupied by the problems of finding conditions for stability and stabilization in the sense of N.E. Zhukovsky of trajectories of dynamic systems modeled by systems of three nonlinear differential equations of the first order. The aim of the work is to study the stability in the Zhukovsky sense of the trajectories of a dynamic system described by three autonomous nonlinear first-order differential equations using a differential geometric method called the accompanying coordinate reference method. The formulation of the stability problem in the Zhukovsky sense of the trajectories of dynamic systems described by systems of three nonlinear differential equations of the first order is considered. The definitions of stability of trajectories of a three-dimensional nonlinear system are specified. Using the accompanying coordinate reference, the conditions of exponential stability and instability according to Zhukovsky are obtained. The results can be used in studying the stability of processes in systems of natural science and technology, as well as in solving theoretical and applied problems of mathematical modeling and qualitative research of trajectories of nonlinear dynamic systems.

Masina O.N., Druzhinina O.V., Lisovsky E.V. Investigation of the stability of trajectories for three-dimensional nonlinear dynamic system using an accompanying coordinate reference. Nonlinear World. 2024. V. 22. № 2. P. 64-69. DOI: https://doi.org/10.18127/ j20700970-202402-08 (In Russian)

1. Zhukovskij N.E. O prochnosti dvizheniya. Uchenye zapiski Moskovskogo universiteta. Otd. fiz.-mat. 1882. Vyp. 4. S. 1–104 (In Russian).
2. Galiullin A.S., Shestakov A.A. Ustojchivost' dvizheniya i variacionnye principy dinamiki. Vestnik RUDN. Ser. Prikladnaya matematika i informatika. 1996. № 2. S. 20–28 (In Russian).
3. Leonov G.A., Ponomarenko D.V. Kriterii orbital'noj ustojchivosti traektorij dinamicheskih sistem. Izv. vuzov. Ser. Matematika. 1993. № 4. S. 88–94 (In Russian).
4. Leonov G.A. Generalization of the Andronov–Vitt theorem. Regular and Chaotic Dynamics. 2006. V. 11. № 2. P. 281–289.
5. Leonov G.A. Haoticheskaya dinamika i klassicheskaya teoriya ustojchivosti dvizheniya. Moskva–Izhevsk: In t komp'yuternyh issledovanij. 2006 (In Russian).
6. Shiryaev A.S., Husainov R.R., Mamedov Sh.N., Gusev S.V., Kuznecov N.V. O metode G. A. Leonova vychisleniya linearizacii transversal'noj dinamiki i analiza ustojchivosti po Zhukovskomu. Vestnik Sankt-Peterburgskogo universiteta. Matematika. Mekhanika. Astronomiya. 2019. T. 6 (64). Vyp. 4. S. 544–554 (In Russian).
7. Druzhinina O.V., Shestakov A.A. Ob usloviyah prochnosti v smysle Zhukovskogo traektorij dinamicheskih sistem. DAN. 2003. T. 393. № 4. S. 478–482 (In Russian).
8. Druzhinina O.V., Shestakov A.A. Vzaimosvyaz' ustojchivosti po Zhukovskomu s ponyatiyami ustojchivosti topologicheskoj dinamiki. Nelinejnyj mir. 2013. T. 11. № 7. S. 459–467 (In Russian).
9. Ding C., Soriano J.M. Uniformly asymptotically Zhukovskij stable orbits. Computers Math. Appl. 2005. V. 49. P. 81–84.
10. Druzhinina O.V. Ustojchivost' i stabilizaciya po Zhukovskomu dinamicheskih sistem. M.: Izdatel'skaya gruppa URSS. 2013 (In Russian).
11. Stepanov V.V. Ob ustojchivosti po Yakobi. Astronomicheskij zhurnal. 1936. T. 13. № 5. S. 435–449 (In Russian).
12. Sindzh Dzh. Tenzornye metody v dinamike. M.: IL. 1947 (In Russian).
13. Aminov M.Sh. Ob ustojchivosti nekotoryh mekhanicheskih sistem. Trudy Kazanskogo. aviacionnogo instituta. 1949. T. 24. S. 3–69 (In Russian).
14. Pak C.H. Synge’s concept of stability applied to nonlinear normal mode. Non-Linear Mechanics. 2006. V. 41. № 5. P. 657–664.
15. Shestakov A.A., Druzhinina O.V., Masina O.N. Ocenka bezopasnosti dvizheniya rel'sovyh ekipazhej na osnove obobshchennoj tekhnicheskoj ustojchivosti i ustojchivosti po Zhukovskomu. Transport: nauka, tekhnika, upravlenie. 2014. № 2. S. 3–8 (In Russian).
16. Druzhinina O.V., Lisovskij E.V. Issledovanie ustojchivosti po Zhukovskomu traektorij dinamicheskih sistem, modeliruemyh nelinejnymi vektorno-matrichnymi uravneniyami. Nelinejnyj mir. 2019. T. 17. № 2. S. 40–47 (In Russian).
17. Druzhinina O.V., Lisovskij E.V., ShChennikova E.V., Kaledina E.A. Analiz ustojchivosti traektorij dinamicheskih sistem, modeliruemyh nelinejnymi vektorno-matrichnymi differencial'nymi uravneniyami. Nelinejnyj mir. 2020. T. 18. № 4. S. 5–14 (In Russian).
18. Druzhinina O.V., Lisovskij E.V. Analiz ustojchivosti traektorij trekhmernyh nelinejnyh dinamicheskih sistem. Nelinejnyj mir. 2023. T. 21. № 2. S. 69–75 (In Russian).