## 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)

Zbl 0747.47010
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### References:

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