Converse of Lyapunov's theorem on asymptotic stability to the first approximation for \(\mathbb{C}\)-analytic nonautonomous systems (Q1907392)

Consider the system (1) \(dz/dt= f(t, z)\), where \(f: \mathbb{R}\times W\to \mathbb{C}^n\) is a continuous function holomorphic in the second argument, \(0\in W\subset \mathbb{C}^n\), \(f(t, 0)= 0\). Along with (1) is considered the variational equation (2) \(du/dt= {\mathcal A}(t)u\), in which \({\mathcal A}(t)= {\partial f\over \partial z} (t, 0)\). It is proved that if the zero solution of (1) is stable (uniformly stable), then the zero solution of (2) is stable (uniformly stable); if the zero solution of (1) is attracting (uniformly attracting), then the zero solution of (2) is attracting (uniformly attracting). Moreover, if the zero solution of (1) is uniformly asymptotically stable then the domain of attraction of the zero solution is unbounded.