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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.

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.

Reviewer: Serguei Zelik (Moskva)

### 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) |

### Keywords:

dynamical systems; stability of boundary equilibria; cosymmetry of vector fields; converse implicit function theorem### Citations:

Zbl 0747.47010
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\textit{L. G. Kurakin}, Math. Notes 63, No. 4, 503--508 (1998; Zbl 0953.34043); translation from Mat. Zametki 63, No. 4, 572--578 (1998)

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### References:

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