BIFURCATIONS OF ONE-PARAMETER FAMILIES OF STEADY STATE REGIMES IN MODEL OF A FILTRATIONAL CONVECTION
Cite this article as:
Govorukhin V. N. BIFURCATIONS OF ONE-PARAMETER FAMILIES OF STEADY STATE REGIMES IN MODEL OF A FILTRATIONAL CONVECTION. Izvestiya VUZ. Applied Nonlinear Dynamics, 2012, vol. 20, iss. 6, pp. 3-14. DOI: https://doi.org/10.18500/0869-6632-2012-20-6-3-14
Results of numerical investigation of bifurcations of one-parameter families of steady state regimes in a planar filtrational convection problem are presented. Galerkin’s method is applied for approximation of partial differential equations. As a result of the cosymmetry existence there are curves of equilibria with the hidden parameter. The algorithm of calculation of such curves is described. This algorithm can be applied to analyze systems with nonisolated sets of equilibria. The following bifurcations of equilibria curves are found: emergence of family of equilibriums on already existing family, subdivision of family of equilibria, emergence of family of equilibria «from air», crossing of families of equilibriums and existence of the composite equilibria sets.
1. Юдович В.И. Косимметрия, вырождение решений операторных уравнений, возникновение фильтрационной конвекции // Математические заметки. 1991. Т. 49, № 5. С. 142.
2. Yudovich V.I. Secondary cycle of equilibria in a system with cosymmetry, its creation by bifurcation and impossibility of symmetric treatment of it // Chaos. 1995. Vol. 5, № 2. P. 402.
3. Юдович В.И. Теорема о неявной функции для косимметричных уравнений // Математические заметки. 1996. Т. 60, № 2. С. 313.
4. Kurakin L.G., Yudovich V.I. Bifurcations accompanying monotonic instability of an equilibrium of a cosymmetric dynamical system // Chaos. 2000. Vol. 10,№2. P. 311.
5. Любимов Д.В. О конвективных движениях в пористой среде, подогреваемой снизу // ПМТФ. 1975. № 2. С. 131.
6. Говорухин В.Н. Численное исследование потери устойчивости вторичными стационарными режимами в задаче плоской конвекции Дарси // ДАН. 1998. Т. 363, № 6. С. 752.
7. Говорухин В.Н. Анализ семейств вторичных стационарных режимов в задаче плоской фильтрационной конвекции в прямоугольном контейнере // Изв. РАН. МЖГ. 1999. № 5. С. 53.
8. Говорухин В.Н., Шевченко И.В. Численное исследование второго перехода в задаче плоской фильтрацилнной конвекции // Изв. РАН. МЖГ. 2003. № 5. С. 115.
9. Karas¨ozen B., Tsybulin V. Finite difference approximations and cosymmetry conservation in filtration-convection problem // Phys. Lett. A. 1999. Vol. 262, № 4–5. P. 321.
10. Karas¨ozen B., Tsybulin V. Cosymmetric families of steady states in Darcy convection and their collision // Phys. Lett. A. 2004. Vol. 323, № 1–2. P. 67.
11. Говорухин В.Н., Шевченко И.В. Сценарии возникновения нестационарных режимов в задаче плоской фильтрационной конвекции // Изв. РАН. МЖГ. 2006. № 6. С. 125.
12. Govorukhin V. Calculation of one-parameter families of stationary regimes in a cosymmetric case and analysis of plane filtrational convection problem // Continuation methods in fluid dynamics. Notes Numer. Fluid Mech. 74. Braunschweig: Vieweg, 2000. P. 133.
13. Govorukhin V. Computer experiments with cosymmetric models // Z. Angew. Math. Mech. 1996. Vol. 76. P. 559.
14. Allgower E.L., Georg K. Introduction to numerical continuation methods. Reprint of the 1979 original. Philadelphia, PA: SIAM Society for Industrial and Applied Mathematics, 2003.
15. Kuznetsov Y.A. Elements of applied bifurcation theory. 3rd ed. New York: Springer, 2004.
16. Кузнецов Е., Шалашилин В. Наилучший параметр продолжения решения // Доклады академии наук. 1994. Т. 334, № 5. С. 566.
17. Рикс Е. Применение метода Ньютона к задаче упругой устойчивости // Прикл. механ. 1972. № 4. С. 204.
18. Ворович И.И., Зипалова В.Ф. К решению нелинейных краевых задач теории упругости методом перехода к задаче Коши // Прикл. матем. и механ. 1965. Т. 29, № 5. С. 894.
19. Kurakin L., Yudovich V. Bifurcation of the branching of a cycle in n-parameter family of dynamic systems with cosymmetry // Chaos. 1997. Vol. 7, № 3. P. 376.
BibTeX
author = {V N Govorukhin},
title = {BIFURCATIONS OF ONE-PARAMETER FAMILIES OF STEADY STATE REGIMES IN MODEL OF A FILTRATIONAL CONVECTION},
year = {2012},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {20},number = {6},
url = {https://old-andjournal.sgu.ru/en/articles/bifurcations-of-one-parameter-families-of-steady-state-regimes-in-model-of-filtrational},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2012-20-6-3-14},pages = {3--14},issn = {0869-6632},
keywords = {Filtrational convection,cosymmetry,families of equilibria,bifurcations},
abstract = { Results of numerical investigation of bifurcations of one-parameter families of steady state regimes in a planar filtrational convection problem are presented. Galerkin’s method is applied for approximation of partial differential equations. As a result of the cosymmetry existence there are curves of equilibria with the hidden parameter. The algorithm of calculation of such curves is described. This algorithm can be applied to analyze systems with nonisolated sets of equilibria. The following bifurcations of equilibria curves are found: emergence of family of equilibriums on already existing family, subdivision of family of equilibria, emergence of family of equilibria «from air», crossing of families of equilibriums and existence of the composite equilibria sets. }}