Construction and localization of quasiinvariant sets of dynamical systems (Q1874840)

The paper deals with a system of differential equations \[ \dot x = f(x), x \in \mathbb{R}^n, \tag{1} \] where the vector-function \(f\) satisfies the local Lipschitz condition and origin \(0 \in \mathbb{R}^n\) is the unique singular point of (1). A connected set \(G \subset \mathbb{R}^n\) is said to be quasi-invariant with respect to (1) if solutions \(x(t, x^0) \in G\) for all \(t \geq t_0\) and all \(x^0 \in G,\) \(x(t_0, x^0) = x^0.\) The author constructs quasiinvariant sets of (1) using level surfaces of some functions, including linear functions for which the construction is most effective. Using a comparison system, solutions of (1) on quasiinvariant sets are estimated.