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

Grishanina, G. É.; Inozemtseva, N. G.; Sadovnikov, B. I.

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 \([0, T]\)”.

Reviewer: Daniela Danciu (Craiova)

MSC:

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

Keywords:

first-order nonlinear system; global asymptotic stability; stability of saddle solutions

Citations:

Zbl 0089.29502

References:

[1]	Н. Н. Боголюбов, Ю. А. Митропольский, Асимптотические методы в теории нелинейных колебаний, Наука, М., 1974 · Zbl 0303.34043
[2]	А. В. Борисов, В. В. Козлов, И. С. Мамаев, Нелинейная динам., 3:3 (2007), 255 – 296
[3]	A. V. Borisov, V. V. Kozlov, I. S. Mamaev, Regul. Chaotic Dyn., 12:5 (2007), 531 – 565 · Zbl 1229.37107
[4]	Е. А. Барабашин, Н. Н. Красовский, ПММ, 18:3 (1954), 445 – 450
[5]	И. Г. Малкин, Теория устойчивости движения, Едиториал УРСС, М., 2010 · Zbl 0136.08502
[6]	Н. Лузин, Матем. сб., 39:3 (1932), 6 – 26 · Zbl 0006.11404
[7]	Г. Э. Гришанина
[8]	В. В. Немыцкий, В. В. Степанов, Качественная теория дифференциальных уравнений, М., Гостехиздат, 1949