## On global asymptotic stability and stability of saddle solutions at infinity.(English)Zbl 1280.34058

Math. Notes 93, No. 4, 624-628 (2013); translation from Mat. Zametki 93, No. 4, 624-629 (2013).
The authors consider the system $\dot{x}=F(x)$ with $$\dim x = n$$ and $$F: \mathbb{R}^n \rightarrow \mathbb{R}^n$$ satisfying the existence and uniqueness conditions. They study the connection between asymptotic stability of a given equilibrium and the property of a saddle at infinity in the sense of V. V. Nemytskii and V. V. Stepanov [Qualitative theory of differential equations. New Jersey: Princeton University Press (1960; Zbl 0089.29502)]. Several theorems are proved. Among them, the most important result is Theorem 4 claiming that for the nonexistence of a saddle at infinity it is necessary and sufficient that there exists “a nondecreasing function $$\sigma: [0, \infty) \rightarrow [0, \infty)$$ such that $$\|x(t)\| \leq \sigma(\|x(0)+x(t)\|)$$, $$0 \leq t \leq T$$, for any solution $$x(t)$$ defined on the closed interval $$[0, T]$$”.

### MSC:

 34D23 Global stability of solutions to ordinary differential equations 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

Zbl 0089.29502
Full Text:

### References:

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