Perron stability and its study at the first approximation (Q2332079)

For a system of ordinary differential equations of the form \[ \dot{x}=f(t,x), \ \ \ f(t,0)\equiv0, \ \ (t,x)\in\mathbb{R}^+\times G,\tag{1} \] where \(\mathbb{R}^+=[0,\infty)\), \(G\) is a neighbourhood of the point \(0\), \(f, f_x\in C(\mathbb{R}^+\times G)\). Let \(\mathscr{S}_{\delta}(f)\) be the set of all nonextendable nontrivial solutions \(x(t)\) of the system such that \(\|x(0)\|<\delta\). The author introduces the following definition: the trivial solution \(x=0\) of System (1) is said to be \begin{itemize} \item[(i)] Perron stable if \(\exists\,\varepsilon>0\) \(\forall\delta>0\) such that any solution \(\forall x\in\mathscr{S}_{\delta}(f)\) \(\underline{\lim}_{t\to\infty}\|x(t)\|<\varepsilon\); \item[(ii)] Perron asymptotically stable if \(\exists\,\delta>0\) such that any solution \(\forall x\in\mathscr{S}_{\delta}(f)\) \(\underline{\lim}_{t\to\infty}\|x(t)\|=0\); \item[(iii)] Perron unstable if \(\forall\varepsilon>0\) \(\exists\delta>0\) such that at least one solution \(x\in\mathscr{S}_{\delta}(f)\) does not satisfy the condition \(\underline{\lim}_{t\to\infty}\|x(t)\|<\varepsilon\); \item[(iv)] Perron completely unstable if \(\exists\,\varepsilon>0\), \(\delta>0\) such that any solution \(x\in\mathscr{S}_{\delta}(f)\) does not satisfy the condition \(\underline{\lim}_{t\to\infty}\|x(t)\|<\varepsilon\). \end{itemize} This definition is associated with the Perron exponent (see [\textit{O. Perron}, Math. Z. 31, 748--766 (1930; JFM 56.1038.03)]). The new concept of Perron stability is compared with the classical Lyapunov stability properties. An absolute and unique coincidence of research opportunities for Perron and Lyapunov stability and asymptotic stability at the first approximation is found. For more details and further study, see the author's other papers [``Definition of Perron stability and its relation to Lyapunov stability'', Differ. Uravn. 54, No. 6, 855--856 (2018); ``On the study on the Perron stability of one-dimensional and autonomous differential systems'', ibid. 54, No. 11, 1561--1562 (2018); Differ. Equ. 55, No. 5, 620--630 (2019; Zbl 1423.34072); translation from Differ. Uravn. 55, No. 5, 636--646 (2019); ``On the study of Perron and Lyapunov stability properties in the first approximation'', Differ. Uravn. 55, No. 6, 897--899 (2019); Differ. Equ. 55, No. 10, 1294--1303 (2019; Zbl 1440.93196); translation from Differ. Uravn. 55, No. 10, 1338--1346 (2019)].