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Instability of isolated equilibria of dynamical systems with invariant measure in spaces of odd dimension. (English. Russian original) Zbl 0962.34036
Math. Notes 65, No. 5, 565-570 (1999); translation from Mat. Zametki 65, No. 5, 674-680 (1999).
Consider an autonomous system of differential equations in the space \({\mathbb R}^n\), where \(n\) is an odd number. It is assumed that the system has an invariant measure with smooth density and that \(x=0\) is an isolated rest point. It was conjectured by V. V. Ten [Mosc. Univ. Mech. Bull. 52, No. 3, 1-4 (1997); translation from Vestn. Mosk. Univ., Ser. I 1997, No. 3, 40-43 (1997; Zbl 0916.34050)] that in this case, the rest point \(x=0\) is unstable. The authors prove this conjecture in the case where the system has a quasihomogeneous truncation for which \(x=0\) is an isolated rest point. They also construct an example of a system of the class \(C^\infty\) for which the conjecture fails.

MSC:
34D20 Stability of solutions to ordinary differential equations
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