Asymptotic stability of linearizations of a planar vector field with a singular point implies global stability
The author presents a proof of a result already announced in [Usp. Mat. Nauk 49, No. 3, 179-180 (1994; Zbl 0828.34041)] which gives a positive solution to the famous ``global asymptotic stability Jacobian conjecture'' [cf. \textit{G. Meisters} and \textit{C. Olech}, Analyse mathématique et applications, Contrib. Honneur Jacques-Louis Lions, 373-381 (1988; Zbl 0668.34048)]) for \(n = 2\) and reads as follows: Theorem. Let a vector field in \(\mathbb{R}^2\) have a singular point, and let its linearization be asymptotically stable at every point of the phase space. Then the singular point is unique and globally attractive.
global asymptotic stability Jacobian conjecture
scientific article; zbMATH DE number 923586
scientific article