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Critical cases of stability. Converse implicit function theorem for dynamical systems with cosymmetry. (English. Russian original) Zbl 0953.34043

Math. Notes 63, No. 4, 503-508 (1998); translation from Mat. Zametki 63, No. 4, 572-578 (1998).
The author studies the stability for the equilibria point \(u_0\in\mathbb{R}^n\) of ODEs \[ \dot u=F(u),\;u\in\mathbb{R}^n,\;\;F(u_0)=0,\tag{1} \] \(F\) is analytic which admits a nontrivial cosymmetry, i.e., it is assumed that there is a continuous vector field \(L\), such that \[ (F(u),L(u))\equiv 0,\quad L(u_0)\neq 0. \tag{2} \] It is known (see e.g. V. I. Yudovich [Math. Notes 49, No. 5, 540-545 (1991); translation from Mat. Zametki 49, No. 5, 142-148 (1991; Zbl 0747.47010)] that under assumption (2) equation (1) generically has a one-parametrical analytic curve of equilibrium points which contains \(u_0\).
The author proves that the converse assertion is also true: namely, if the equation has a one-parametrical analytic curve of equilibrium points (and satisfies some natural assumptions) then it possesses a nontrivial cosymmetry (2).
Moreover, criteria for the stability of the equilibria for the systems of ODEs with cosymmetry in all critical (boundary) cases of codimension \(0\) and \(1\) are also presented.

MSC:

34D20 Stability of solutions to ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
37C75 Stability theory for smooth dynamical systems
37C80 Symmetries, equivariant dynamical systems (MSC2010)

Citations:

Zbl 0747.47010
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References:

[1] V. I. Yudovich, ”Cosymmetry, the degeneration of solutions of operator equations, the initiation of filtrational convection,”Mat. Zametki [Math. Notes],49, No. 5, 142–148 (1991).
[2] V. I. Yudovich, ”Cosymmetry and differential equations of the second order,” in:Dep. VINITI No.1008-93 [in Russian], VINITI, Moscow (1993).
[3] V. I. Yudovich, ”Secondary cycle of equilibria in a system with cosymmetry, its creation by bifurcation and impossibility of symmetric treatment of it,”Chaos,2, No. 5, 402–411 (1995). · Zbl 1055.58500
[4] V. I. Yudovich, ”Implicit function theorem for cosymmetric equations,”Mat. Zametki [Math. Notes],60, No. 2, 313–317 (1996). · Zbl 0897.47051
[5] V. I. Yudovich, ”The cosymmetric version of the implicit function theorem,” in:Seminar ”Linear Topological Spaces and Complex Analysis.” II (A. Aytuna, editor), Middle East Technical Univ., Ankara (1995), pp. 105–125. · Zbl 0860.46030
[6] A. M. Lyapunov,General Problem of the Stability of Motion [in Russian], Gostekhizdat, Moscow (1950). · Zbl 0041.32204
[7] A. M. Lyapunov, ”The study of one of the singular cases in the problem about the stability of motion,” in:Collected Works [in Russian], Vol. 2, AN SSSR, Moscow (1956), pp. 272–331.
[8] L. G. Khazin,Conditions of Equilibrium Stability for Two Pairs of Pure Imaginary and One Zero Root [in Russian], Preprint No.50, Inst. of Appl. Math., Moscow (1986).
[9] V. A. Pliss, ”Reduction principle in the theory of stability of motion,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],28, No. 6, 1297–1324 (1964). · Zbl 0131.31505
[10] L. G. Khazin and É. É. Shnol’,Stability of Critical Equilibria [in Russian], Sci. Tech. Inf. Dept. of the Sci. Center for Biol. Research of Akad. Nauk SSSR, Pushchino (1985). · Zbl 0960.34510
[11] V. I. Arnold,Supplementary Chapters of the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1978).
[12] É. É. Shnol’ and L. G. Khazin,On Stability of Stationary Solutions to General Systems of Differential Equations Near Critical Cases [in Russian], Preprint No.91, Inst. of Appl. Math., Moscow (1979). · Zbl 0416.34053
[13] V. V. Rumyantsev, ”On stability of motion with respect to a part of the variables,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], No. 4, 9–16 (1957).
[14] G. V. Kamenkov,Selected Works. Vol. 2. Stability and Oscillations of Nonlinear Systems, Nauka, Moscow (1972).
[15] I. G. Malkin,Theory of Motion Stability, Nauka, Moscow (1966). · Zbl 0136.08502
[16] L. G. Khazin,On Stability of Equilibria in Certain Critical Cases, Preprint No.10, Inst. of Appl. Math., Moscow (1979). · Zbl 0416.34053
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