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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: 10.18500/0869-6632-2012-20-6-3-14

# Bifurcations of one-parameter families of steady state regimes in model of a filtrational convection

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.

- Yudovich VI. Cosymmetry, degeneration of solutions of operator equations, and onset of a filtration convection. Mathematical Notes of the Academy of Sciences of the USSR. 1991;49(5):540–545. DOI: https://doi.org/10.1007/BF01142654.
- Yudovich VI. Secondary cycle of equilibria in a system with cosymmetry, its creation by bifurcation and impossibility of symmetric treatment of it. Chaos. 1995;5(2):402–411. DOI: 10.1063/1.166110.
- Yudovich VI. Implicit function theorem for cosymmetric equations. Math. Notes. 1996;60(2):235–238. DOI: 10.1007/BF02305191.
- Kurakin LG, Yudovich VI. Bifurcations accompanying monotonic instability of an equilibrium of a cosymmetric dynamical system. Chaos. 2000;10(2):311–330. DOI: 10.1063/1.166497.
- Lyubimov DV. Convective motions in a porous medium heated from below. J. Appl. Mech. Tech. Phys. 1975;16(2):257–261. DOI: 10.1007/BF00858924.
- Govorukhin VN. Numerical study of the loss of stability by secondary stationary regimes in the plane Darcy convection problem. Proc. Acad. Sci. USSR. 1998;363(6):772–773 (in Russian).
- Govorukhin VN. Analysis of families of secondary stationary regimes in the problem of plane filtration convection in a rectangular container. Fluid Dynamics. 1999;(5):53–62 (in Russian).
- Govorukhin VN, Shevchenko IV. Numerical investigation of the second transition in the problem of plane convective flow through a porous medium. Fluid Dynamics. 2003;38(5):760–771. DOI: 10.1023/B:FLUI.0000007838.46669.1a.
- Karasozen B, Tsybulin V. Finite difference approximations and cosymmetry conservation in filtration-convection problem. Phys. Lett. A. 1999;262(4–5):321–329. DOI: 10.1016/S0375-9601(99)00599-X.
- Karasozen B, Tsybulin V. Cosymmetric families of steady states in Darcy convection and their collision. Phys. Lett. A. 2004;323(1–2):67–76. DOI: 10.1016/j.physleta.2004.01.053.
- Govorukhin VN, Shevchenko IV. Scenarios of the onset of unsteady regimes in the problem of plane convective flow through a porous medium. Fluid Dynamics. 2006;41(6):967–975. DOI: 10.1007/s10697-006-0111-2.
- Govorukhin V. Calculation of one-parameter families of stationary regimes in a cosymmetric case and analysis of plane filtrational convection problem. In: Continuation Methods in Fluid Dynamics. Notes Numer. Fluid Mech. 74. Braunschweig: Vieweg; 2000. P. 133–144.
- Govorukhin V. Computer experiments with cosymmetric models. Z. Angew. Math. Mech. 1996;76(4):559–562.
- Allgower EL, Georg K. Introduction to Numerical Continuation Methods. Reprint of the 1979 Original. Philadelphia, PA: SIAM Society for Industrial and Applied Mathematics; 2003. 388 p. DOI: 10.1137/1.9780898719154.
- Kuznetsov YA. Elements of Applied Bifurcation Theory. 3rd ed. New York: Springer; 2004. 632 p. DOI: 10.1007/978-1-4757-3978-7.
- Kuznetsov E, Shalashilin V. The best solution continuation parameter. Doklady Mathematics. 1994;49(1):170–173.
- Ricks E. Application of Newton's method to the problem of elastic stability. Applied Mechanics. 1972;(4):204–210 (in Russian).
- Vorovich II, Zipalova VF. On the solution of nonlinear boundary value problems of the theory of elasticity by a method of transformation to an initial value. Journal of Applied Mathematics and Mechanics. 1965;29(5):1055–1063.
- Kurakin L, Yudovich V. Bifurcation of the branching of a cycle in n-parameter family of dynamic systems with cosymmetry. Chaos. 1997;7(3):376–386. DOI: 10.1063/1.166250.

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