Global stability of two-dimensional systems (Q5942183)

Here, the global asymptotic stability of solutions to systems of differential equations of the form \[ dx/dt=f(x), \tag{1} \] with \(x(t)\in\mathbb{R}^2\), \(f \in C^1\) \((\mathbb{R}^2 \to\mathbb{R}^2)\) and \(f(0)=0\), is studied. This problem was studied by \textit{C. Olech} [Contrib. Differ. Equations 1, 389-400 (1963; Zbl 0136.08602)] when \(f'(x)>0\), \(\text{div} f(x)<0\) \(\forall x\in\mathbb{R}^2\). The goal of this work is to study this problem under less restrictive assumptions on \(f\). A number of results are represented. So, if \(\|f(x)\|> \varepsilon /(1+\|x\|)>0\) for all \(x\in \mathbb{R}^2\) outside some ball centered at the origin, then system (1) is globally asymptotically stable. Let \(O(\varepsilon, \mathbb{R})=\{x: \|f(x) \|<\varepsilon\), \(\|x\|> \mathbb{R}\}\). Suppose that for any positive numbers \(\varepsilon\) and \(R_0\), there exists a number \(R>R_0\) and a closed curve \(\gamma=\{z(t):0\leq t\leq T,z(0)=z (T),z \in C^1\}\) without self-intersections such that this curve surrounds the origin and lies outside the disk of radius \(R\) centered at the origin and \((dx(t)/dt, \partial f(z(t))) \neq 0\) for any \(z(t)\in\gamma\cap O(\varepsilon, R)\). Then system (1) is globally asymptotically stable.