On the stability of gradient polynomial systems at infinity (Q603036)

Consider the differential equation \[ \dot{x}(t)=-\nabla f(x(t)), \tag{1} \] where \(f:\mathbb R^{n}\to\mathbb R\) is a real continuously differentiable function and \(\nabla f(x)\) denotes the gradient of \(f\) at \(x\). Definition 1. Let \(f\) be a real polynomial. A value \(y_{0}\in\mathbb R\) is called a local infimum at infinity of \(f\) if the following two conditions hold {\parindent6mm \begin{itemize}\item[(i)] there exist \(\delta,r >0\) such that \(\|x\|\geq r\) and \(|f(x) -y_{0}|< \delta \Longrightarrow f(x)\geq y_{0};\) \item[(ii)] there exists a sequence \(x^{k}\to\infty\) such that \(f(x^{k})\to y_{0}.\) \end{itemize}} Additionally, if \(\delta\) and \(r\) can be chosen such that \[ \|x\| \geq r\,\, \text{and}\,\, |f(x) - y_0| <\delta \Longrightarrow f(x)> y_{0} \] then \(y_{0}\) is called an isolated infimum value at infinity of \(f.\) Definition 2. Let \(f:\mathbb R^{n}\to\mathbb R, n>1, \) be a non-constant polynomial and \(y_{0}\in\mathbb R.\) System (1) is called stable at infinity near the fiber \(f^{-1}(y_{0})\) if for every \(r >0\), \(\varepsilon >0\) there exist \(\delta >0\), \(r_{0} >0\) such that \[ \|x(0)\| > r_{0}\,\, \text{and}\,\, |f(x(0))-y_{0}|<\delta \Longrightarrow \|x(t)\|> r\,\, \text{and}\,\, |f(x(t))-y_{0}|<\varepsilon\,\, \forall t \geq 0. \] It is asymptotically stable at infinity near the fiber \(f^{-1}(y_{0})\) if it is stable and \(\delta\), \(r_{0}\) can be chosen such that \[ \|x(0)\|>r_{0}\,\,\text{and}\,\,|f(x(0))-y_{0}|<\delta \Longrightarrow \lim_{t\to\infty}x(t)=\infty\,\, \text{and}\,\, \lim_{t\to\infty}f(x(t))= y_{0}, \] Theorem. Let \(f:\mathbb R^{n}\to\mathbb R\) be a polynomial and \(y_{0}\in\mathbb R\) be an isolated infimum value at infinity of \(f\). Then (1) is asymptotically stable at infinity near the fiber \(f^{-1}(y_{0}).\)