Abstract
Using the direct Lyapunov method, we study the stability of an equilibrium of a cosymmetric vector field in the case when the stability spectrum lies in the closure of the left half-plane and the neutral spectrum (lying on the imaginary axis) consists of simple eigenvalues zero and a pair of purely imaginary numbers. Owing to cosymmetry, this equilibrium state is a member of a continuous one-parameter family of equilibria with a variable stability spectrum. We use theorems on asymptotic stability with respect to part of variables. We find stability criteria in the case of general position, as well for all degenerations of codimension one and one case of codimension two. As a result, we give description for dangerous and safe stability boundaries.
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Yudovich V. I., “Cosymmetry, degeneration of solutions of operator equations, and onset of filtration convection,” Mat.Zametki, 49, No. 5, 142–148 (1991).
Yudovich V. I., “Secondary cycle of equilibria in a system with cosymmetry, its creation by bifurcation and impossibility of symmetric treatment of it,” Chaos, 5, No. 2, 402–411 (1995).
Yudovich V. I., “Implicit function theorem for cosymmetric equations,” Mat. Zametki, 60, No. 2, 313–317 (1996).
Yudovich V. I., “Cosymmetric version of implicit function theorem,” in: Proceedings of “Linear Topological Spaces and Complex Analysis,” Middle East Technical Univ., Ankara, 1995, 2, p. 16.
Yudovich V. I., “Cosymmetry and dynamical systems,” in: Proc. ICIAM 95, 4: Applied Sciences, Especially Mechanics, pp. 585–588.
Lyapunov A. M., The General Problem of the Stability of Motion [in Russian], Gostekhizdat, Moscow (1950).
Bautin N. N., Behavior of Dynamical Systems Near a Boundary of Stability Domain [in Russian], Nauka, Moscow and Leningrad (1984).
Khazin L. G. and Shnol 0 E. E., Stability of Critical Equilibrium States [in Russian], ONTI NTsBI Akad. Nauk SSSR, Pushchino (1985).
Lyapunov A. M., Study of One of Singular Cases of the Stability of Motion. Vol. 2 [in Russian], Izdat. Akad. Nauk SSSR, Moscow, 1956, pp. 272–331.
Rumyantsev V. V., “On the Stability of Motion with Respect to Part of Variables,” Vestnik Moskovsk. Univ. Ser. 1 Mat. Mekh., No. 4, 9–16 (1957).
Yudovich V. I., “Cycle-creating bifurcation from a family of equilibria of a dynamical system and its contraction,” Prikl.Mat. Mekh., 62, No. 1, 22–34 (1998).
Kurakin L. G. and Yudovich V. I., “Bifurcation of the branching of a cycle in n-parameter family of dynamic system with cosymmetry,” Chaos, 7, No. 3, 376–386 (1997).
Kurakin L. G. and Yudovich V. I., “Cycle-creating bifurcation in a system with cosymmetry,” Dokl. Akad. Nauk, 358, No. 3, 346–349 (1998).
Kurakin L. G. and Yudovich V. I., “Branching of a limit cycle from the equilibrium submanifold in a system with multicosymmetries,” Mat. Zametki, 66, No. 2, 317–320 (1999).
Kurakin L. G. and Yudovich V. I., “Application of the Lyapunov-Schmidt method to the problem of the branching of a cycle from a family of equilibria in a system with multicosymmetry,” Sibirsk. Mat. Zh., 41, No. 1, 136–149 (2000).
Kamenkov G. V., Stability and Oscillations of Nonlinear Systems. Selected Works. Vol. 2 [in Russian], Nauka, Moscow (1972).
Malkin I. G., A Theory of Stability of Motion [in Russian], Nauka, Moscow (1966).
Khazin L. G., On Stability of Equilibrium States in Some Critical Cases [in Russian] [Preprint, No. 242 10], Inst. Prikl.Mat. Akad. Nauk SSSR, Moscow (1979).
Kurakin L. G., “Critical cases of stability. A converse of the implicit function theorem for dynamical systems with cosymmetry,” Mat. Zametki, 63, No. 4, 572–578 (1998).
Pliss V. A., “The reduction principle in the theory of stability of motion,” Izv. Akad. Nauk Ser. Mat., 6, No. 6, 1297–1324 (1964).
Arnol' d V. I., Supplementary Chapters to the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1978).
Kurakin L. G. and Yudovich V. I., “Semi-invariant form of equilibrium stability criteria in critical cases,” Prikl. Mat.Mekh., 50, No. 5, 707–711 (1986).
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Kurakin, L.G. On Stability of Boundary Equilibria in Systems with Cosymmetry. Siberian Mathematical Journal 42, 1102–1110 (2001). https://doi.org/10.1023/A:1012844610886
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DOI: https://doi.org/10.1023/A:1012844610886