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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

Citations:

Zbl 0089.29502
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References:

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