GROUP-THEORETICAL METHODS FOR SIMPLIFICATION OF STABILITY ANALYSIS OF DYNAMICAL REGIMES IN NONLINEAR SYSTEMS WITH DISCRETE SYMMETRY
Cite this article as:
Zhukov К. G., Chechin G. М. GROUP-THEORETICAL METHODS FOR SIMPLIFICATION OF STABILITY ANALYSIS OF DYNAMICAL REGIMES IN NONLINEAR SYSTEMS WITH DISCRETE SYMMETRY. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, iss. 4, pp. 147-166. DOI: https://doi.org/10.18500/0869-6632-2008-16-4-147-166
We present a detailed description of the group-theoretical method which has been published in 2006 by the authors. This method can frequently simplify the study of the stability of different dynamical regimes in nonlinear physical systems with discrete symmetry since it allows one to split the set of the linearized (near a considered regime) nonlinear differential equations into a number of independent subsets of small dimensions. The above method is illustrated with the case of stability analysis of some dynamical regimes in the simple octahedral structure.
1. Chechin G. M., Zhukov K. G. Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries // Phys. Rev. E. 2006. Vol. 73. P. 36216.
2. Сахненко В. П., Чечин Г. М. Симметрийные правила отбора в нелинейной динамике атомных систем // ДАН. 1993. Т. 330. С. 308.
3. Сахненко В. П., Чечин Г. М. Кусты мод и нормальные колебания для нелинейных динамических систем с дискретной симметрией // ДАН. 1994. Т. 338. С. 42.
4. Chechin G. M., Sakhnenko V. P. Interactions between normal modes in nonlinear dynamical systems with discrete symmetry. Exact results // Physica D. 1998. Vol. 117. P. 43.
5. Chechin G. M., Ryabov D. S., Sakhnenko V. P. Bushes of normal modes as exact excitations in nonlinear dynamical systems with discrete symmetry. In: «Nonlinear Phenomena Research Perspectives», p. 225, ed. C.W. Wang, Nova Science Publishers, NY, 2007.
6. СиротинЮ. И., Шаскольская М. П. Основы кристаллофизики. М.: Наука, 1975.
7. Ландау Л. Д., Лифшиц Е. М. Теоретическая физика. Т. I. Механика. М.: Наука, 1988.
8. Rosenberg R. M. The normal modes of nonlinear n-degree-of-freedom systems // J. Appl. Mech. 1962. Vol. 29. P. 7.
9. Rosenberg R. M. On nonlinear vibrations of systems with many degrees of freedom // Adv. Appl. Mech. 1966. Vol. 9. P. 155.
10. Chechin G. M., Novikova N. V., Abramenko A. A. Bushes of vibrational modes for Fermi–Pasta–Ulam chains // Physica D. 2002. Vol. 166. P. 208.
11. Chechin G. M., Ryabov D. S., Zhukov K. G. Stability of low dimensional bushes of vibrational modes in the Fermi–Pasta–Ulam chains//Physica D.2005.Vol. 203. P. 121.
12. Chechin G. M., Sakhnenko V. P., Stokes H. T., Smith A. D., Hatch D. M. Non-linear normal modes for systems with discrete symmetry // Int. J. Non-Linear Mech. 2000. Vol. 35. P. 497.
13. Chechin G. M., Gnezdilov A. V., Zekhtser M. Yu. Existence and stability of bushes of vibrational modes for octahedral mechanical systems with Lennard–Jones potential // Int. J. Non-Linear Mech. 2003. Vol. 38. P. 1451.
14. Эллиот Дж., Добер П. Симметрия в физике. М.: Мир, 1983.
15. Петрашень М. И., Трифонов Е. Д. Применение теории групп в квантовой механике. М.: Наука, 1967.
16. Ландау Л. Д., Лифшиц Е. М. Теоретическая физика. Т. III. Квантовая механика. Нерелятивистская теория. М.: Наука, 1974.
17. Справочник по специальным функциям / Под ред. Абрамовица М. и Стиган И. М.: Наука, 1979.
BibTeX
author = {К. G. Zhukov and G М. Chechin},
title = {GROUP-THEORETICAL METHODS FOR SIMPLIFICATION OF STABILITY ANALYSIS OF DYNAMICAL REGIMES IN NONLINEAR SYSTEMS WITH DISCRETE SYMMETRY},
year = {2008},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {16},number = {4},
url = {https://old-andjournal.sgu.ru/en/articles/group-theoretical-methods-for-simplification-of-stability-analysis-of-dynamical-regimes-in},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2008-16-4-147-166},pages = {147--166},issn = {0869-6632},
keywords = {-},
abstract = {We present a detailed description of the group-theoretical method which has been published in 2006 by the authors. This method can frequently simplify the study of the stability of different dynamical regimes in nonlinear physical systems with discrete symmetry since it allows one to split the set of the linearized (near a considered regime) nonlinear differential equations into a number of independent subsets of small dimensions. The above method is illustrated with the case of stability analysis of some dynamical regimes in the simple octahedral structure. }}