Математика и математическое моделирование. 2015; : 54-65
Локализация инвариантных компактов системы Lorenz-84
Аннотация
Список литературы
1. Канатников А.Н., Крищенко А.П. Инвариантные компакты динамических систем. М.: Изд-во МГТУ им. Н.Э. Баумана, 2011. 231 с.
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4. Крищенко А.П. Локализация инвариантных компактов динамических систем // Дифференциальные уравнения. 2005. Т.41, № 12. С. 1597-1604.
5. Krishchenko A. P., Starkov K.E. Localization of compact invariant sets of the Lorenz system // Phys. Lett. A. 2006. Vol. 353, no. 5. P. 383-388.
6. Li D., Lu J., Wu X., Chen G. Estimating the bounds for the Lorenz family of chaotic systems // Chaos, Solitons and Fractals. 2005. Vol. 23, no. 2. P. 130-141.
7. Канатников А.Н. Локализация инвариантных компактов ПРТ-системы // Вестник МГТУ. Сер. Естественные науки. 2007. №1. С.3-18.
8. Krishchenko A. P., Starkov K.E. Localization of compact invariant sets of nonlinear systems with application to the Lanford systems // Int. J. of Bifurcation and Chaos. 2006. Vol. 16, no. 11. P. 3249-3256.
9. Канатников А.Н., Федорова Ю. П. Локализация инвариантных компактов двумерных непрерывных динамических систем // Наука и образование. МГТУ им. Н.Э. Баумана. Электрон. журн. 2013. № 7. С. 159-174. DOI: 10.7463/0713.0583104.
10. Starkov K. E. Bounding a domain which contains all compact invariant sets of the Bloch system // Int. J. of Bifurcation and Chaos. 2009. Vol. 19, no. 3. P. 1037-1042.
11. Starkov K. E. Bounds for the domain containing all compact invariant sets of system modeling dynamics of acoustic gravity waves // Int. J. of Bifurcation and Chaos. 2009. Vol. 19, no. 10. P. 3425-3432.
12. Lorenz E. N. Irregularity: a fundamental property of the atmosphere // Tellus, 36A (1984), P. 98-110.
13. Masoller C., Schifino A., Romanelli L. Characterization of strange attractors of Lorenz model of general circulation of the atmosphere // Chaos, Solitions and Fractals. 1995. No 6. P. 357-366.
14. Shilnikov A., Nicolis G., Nicolis C. Bifurcation and predictability analysis of a low-order atmospheric circulation model // Journal of Bifurcation and chaos. 1995. No 5. P. 1701-1711.
15. Starkov K. E. Localization of compact invariant sets of the Lorenz’ 1984 model // Springer Proceedings in Physics. 2009. Vol. 132. P. 915.
Mathematics and Mathematical Modeling. 2015; : 54-65
Localization of Compact Invariant Sets of the Lorenz'1984 System
Abstract
Localization of compact invariant sets of a dynamical system is one way to conduct a qualitative analysis of dynamical system. The localization task is aimed at evaluating the location of invariant compact sets of systems, which are equilibrium, periodic trajectories, attractors and repellers, and invariant tori. Such sets and their properties largely determine the structure of the phase portrait of the system. For this purpose, one can use a localization set, i.e. a set in the phase space of the system that contains all invariant compact sets.
This article considers the problem of localization of invariant compact sets of an Autonomous version of the Lorenz-84 system. The system represents a simple model of the General circulation of the atmosphere in middle latitudes. The model was used in various climatological studies. To build localization set of the system the so-called functional localization method is applied. The article describes the main provisions of this method, lists the main properties of the localization sets. The simplest version of the Lorenz-84 system when there are no thermal loads is analyzed, and a common variant of the Autonomous Lorenz-84 system, in which for some values of system parameters chaotic dynamics occurs is investigated. In the first case it is shown that the only invariant compact set of the system is its equilibrium position, and localization function turned out to be a Lyapunov function of the system. For the General version of the system a family of localization sets is built and the intersection of this family is described. Graphical illustration for the localization set at fixed values of the parameters is shown. The result of the study partially overlaps with the result of K.E. Starkov on the subject, but provides additional information.
The theme of localization of invariant compact sets is discussed quite actively in the literature. Research focuses both on the development of the method and its application to dynamical systems of other classes, and on the investigation of specific dynamical systems.
References
1. Kanatnikov A.N., Krishchenko A.P. Invariantnye kompakty dinamicheskikh sistem. M.: Izd-vo MGTU im. N.E. Baumana, 2011. 231 s.
2. Krishchenko A.P. Lokalizatsiya predel'nykh tsiklov // Differentsial'nye uravneniya. 1995. T. 31, № 11. S. 1858-1865.
3. Krishchenko A. P. Oblasti sushchestvovaniya tsiklov // Dokl. RAN. 1997. T. 353, № 1. S. 17-19.
4. Krishchenko A.P. Lokalizatsiya invariantnykh kompaktov dinamicheskikh sistem // Differentsial'nye uravneniya. 2005. T.41, № 12. S. 1597-1604.
5. Krishchenko A. P., Starkov K.E. Localization of compact invariant sets of the Lorenz system // Phys. Lett. A. 2006. Vol. 353, no. 5. P. 383-388.
6. Li D., Lu J., Wu X., Chen G. Estimating the bounds for the Lorenz family of chaotic systems // Chaos, Solitons and Fractals. 2005. Vol. 23, no. 2. P. 130-141.
7. Kanatnikov A.N. Lokalizatsiya invariantnykh kompaktov PRT-sistemy // Vestnik MGTU. Ser. Estestvennye nauki. 2007. №1. S.3-18.
8. Krishchenko A. P., Starkov K.E. Localization of compact invariant sets of nonlinear systems with application to the Lanford systems // Int. J. of Bifurcation and Chaos. 2006. Vol. 16, no. 11. P. 3249-3256.
9. Kanatnikov A.N., Fedorova Yu. P. Lokalizatsiya invariantnykh kompaktov dvumernykh nepreryvnykh dinamicheskikh sistem // Nauka i obrazovanie. MGTU im. N.E. Baumana. Elektron. zhurn. 2013. № 7. S. 159-174. DOI: 10.7463/0713.0583104.
10. Starkov K. E. Bounding a domain which contains all compact invariant sets of the Bloch system // Int. J. of Bifurcation and Chaos. 2009. Vol. 19, no. 3. P. 1037-1042.
11. Starkov K. E. Bounds for the domain containing all compact invariant sets of system modeling dynamics of acoustic gravity waves // Int. J. of Bifurcation and Chaos. 2009. Vol. 19, no. 10. P. 3425-3432.
12. Lorenz E. N. Irregularity: a fundamental property of the atmosphere // Tellus, 36A (1984), P. 98-110.
13. Masoller C., Schifino A., Romanelli L. Characterization of strange attractors of Lorenz model of general circulation of the atmosphere // Chaos, Solitions and Fractals. 1995. No 6. P. 357-366.
14. Shilnikov A., Nicolis G., Nicolis C. Bifurcation and predictability analysis of a low-order atmospheric circulation model // Journal of Bifurcation and chaos. 1995. No 5. P. 1701-1711.
15. Starkov K. E. Localization of compact invariant sets of the Lorenz’ 1984 model // Springer Proceedings in Physics. 2009. Vol. 132. P. 915.
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