Асимптотическое поведение решений систем лоренцевского типа. Аналитические результаты и структуры компьютерных ошибок

Авторы

  • Геннадий Алексеевич Леонов
  • Борис Ростиславич Андриевский
  • Руслан Назирович Мокаев

Аннотация

Для систем лоренцевского типа со сжатием объемов получены аналитические критерии глобальной устойчивости и неустойчивости их стационарных множеств. Описаны и проанализированы компьютерные эксперименты для исследования качественного поведения траекторий систем лоренцевского типа, интерпретация которых без дополнительной проверки, ориентированной на аналитические результаты, может приводить к неверным заключениям. Библиогр. 41 назв. Ил. 5.

Скачивания

Данные скачивания пока недоступны.

Библиографические ссылки

1. Lorenz E.N. Deterministic nonperiodic flow//J. Atmos. Sci. 1963. Vol. 20, N2. P. 130-141.

2. Ruelle D., Takens F. On the nature of turbulence//Communications in mathematical physics. 1971. Vol. 20, N3. P. 167-192.

3. Sparrow C. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. In Ser. Applied Mathematical Sciences. New York: Springer, 1982.

4. Broer H.W., Dumortier F., Van Strien S. J. et al. Structures in dynamics: finite dimensional deterministic studies. Elsevier. 1991. Vol. 2.

5. Sprott J. C. Strange attractors: creating patterns in chaos. Citeseer. 1993.

6. Neimark J.I., Landa P. S. Stochastic and chaotic oscillations. Springer Science & Business Media, 2012. Vol. 77.

7. Hirsch M.W., Smale S., Devaney R. L. Differential equations, dynamical systems, and an introduction to chaos. Academic press, 2012.

8. Shilnikov L.P., Shilnikov A.L., Turaev D.V. et al. Methods of Qualitative Theory in Nonlinear Dynamics: Part 1. World Scientific, 1998.

9. Shilnikov L.P., Shilnikov A.L., Turaev D.V. et al. Methods of Qualitative Theory in Nonlinear Dynamics: Part 2. World Scientific, 2001.

10. Boichenko V.A., Leonov G.A., Reitmann V. Dimension Theory for Ordinary Differential Equations. Stuttgart: Teubner, 2005.

11. Leonov G.A. Strange attractors and classical stability theory. St. Petersburg: St. Petersburg University Press, 2008.

12. Elhadj Z., Sprott J.C. 2-D quadratic maps and 3-D ODE systems: A Rigorous Approach. World Scientific, 2010. Vol. 73.

13. Wiggins S. Global bifurcations and chaos: analytical methods. Springer Science & Business Media, 2013. Vol. 73.

14. Shimada I., Nagashima T. A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems//Progress of Theoretical Physics. 1979. Vol. 61, N6. P. 1605-1616.

15. Doedel E. AUTO: Software for continuation and bifurcation problems in ordinary differential equations. California Institute of Technology, 1986.

16. Parker T.S., Chua L.O. Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag, 1989.

17. Allgower E.L., Georg K. Numerical continuation methods: an introduction. New York: Springer-Verlag, 1990.

18. Dellnitz M., Junge O. Set oriented numerical methods for dynamical systems. In Ser. Handbook of Dynamical Systems. Elsevier Science, 2002. Vol. 2. P. 221-264.

19. Numerical Continuation Methods for Dynamical Systems/Eds B.Krauskopf, H.M.Osinga, J.Galan-Vioque. Dordrecht. The Netherlands: Springer, 2007.

20. Ou Yang S., Wu Y., Lin Y. et al. The discontinuity problem and "chaos" of Lorenz's model//Kybernetes. 1998. Vol. 27, N6/7. P. 621-635.

21. Ou Yang S., Lin Y. Problems with Lorenz's Modeling and the Algorithm of Chaos Doctrine//Frontiers In The Study Of Chaotic Dynamical Systems With Open Problems. World Scientific, 2011. Vol. 16. P. 1-29.

22. Tucker W. The Lorenz attractor exists//Comptes Rendus de l'Acad´emie des Sciences-Series I-Mathematics. 1999. Vol. 328, N12. P. 1197-1202.

23. Viana M. What's new on Lorenz strange attractors?//The Mathematical Intelligencer. 2000. Vol. 22, N3. P. 6-19.

24. Stewart I. Mathematics: The Lorenz attractor exists//Nature. 2000. Vol. 406, N6799. P. 948-949.

25. Leonov G.A. Shilnikov chaos in Lorenz-like systems//International Journal of Bifurcation and

26. Chaos. 2013. Vol. 23, N03. art. num. 1350058. Leonov G.A. Asymptotic integration method for the Lorenz-like system//Doklady Mathematics. 2015. Vol. 462, N5. P. 1-7.

27. Chen G., Ueta T. Yet another chaotic attractor//International Journal of Bifurcation and Chaos. 1999. Vol. 9, N7. P. 1465-1466.

28. Lu J., Chen G. A new chaotic attractor coined//Int. J. Bifurcation and Chaos. 2002. Vol. 12. P. 1789-1812.

29. Tigan G., Opri¸s D. Analysis of a 3D chaotic system//Chaos, Solitons & Fractals. 2008. Vol. 36, N5. P. 1315-1319.

30. Barboza R., Chen G. On the global boundedness of the Chen system//International Journal of Bifurcation and Chaos. 2011. Vol. 21, N11. P. 3373-3385.

31. Zhang F., Liao X., Zhang G. On the global boundedness of the L¨u system//Applied Mathematics and Computation. 2016. Vol. 284. P. 332-339.

32. Zhang F., Mu C., Li X. On the boundness of some solutions of the L¨u system//International Journal of Bifurcation and Chaos. 2012. Vol. 22, N01. P. 1250015.

33. Leonov G.A., Kuznetsov N.V. On differences and similarities in the analysis of Lorenz, Chen, and Lu systems//Applied Mathematics and Computation. 2015. Vol. 256. P. 334-343.

34. Yakubovich V.A., Leonov G.A., Gelig A.Kh. Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities. Singapure: World Scientific, 2004.

35. Leonov G.A., Shumafov M.M. Stabilization of Linear Systems. Cambridge: Cambridge Scientific Publishers, 2012.

36. Zubov N.E., Vorob'eva E.A., Mikrin E.A. et al. Synthesis of stabilizing spacecraft control based on generalized Ackermann's formula//Journ. of Computer and Systems Sciences International. 2011. Vol. 50. P. 93-103.

37. Zubov N.E., Mikrin E.A., Misrikhanov M. Sh. et al. Synthesis of controls for a spacecraft that optimize the pole placement of the close-loop control system//Journ. of Computer and Systems Sciences International. 2012. Vol. 51. P. 431-444.

38. Zubov N.E., Mikrin E.A., Misrikhanov M. Sh. et al. The use of the exact pole placement algorithm for the control of spacecraft motion//Journ. of Computer and Systems Sciences International. 2013. Vol. 52. P. 129-144.

39. Zubov N.E., Mikrin E.A., Misrikhanov M. Sh. et al. Modification of the exact pole placement method and its application for the control of spacecraft motion//Journ. of Computer and Systems Sciences International. 2013. Vol. 52. P. 279-292.

40. Popov V.M. Hyperstability of control systems. Springer Verlag, 1973.

41. LaSalle J. P. Some extensions of Liapunov's second method//IRE Transactions on circuit theory. 1960. Vol. 7, N4. P. 520-527.

Загрузки

Опубликован

20.08.2020

Как цитировать

Леонов, Г. А., Андриевский, Б. Р., & Мокаев, Р. Н. (2020). Асимптотическое поведение решений систем лоренцевского типа. Аналитические результаты и структуры компьютерных ошибок. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия, 4(1), 25–37. извлечено от https://math-mech-astr-journal.spbu.ru/article/view/8572

Выпуск

Раздел

Математика

Наиболее читаемые статьи этого автора (авторов)