Abstract
The article states the sufficient polystability conditions for part of variables for nonlinear systems of ordinary differential equations with a sufficiently smooth right-hand side. The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence. An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed. This operator satisfies the conditions of the Schauder principle, therefore, it has at least one fixed point. Further, using the estimates of the non-zero elements of the fundamental matrix, conditions that ensure the transition of the properties of polystability are obtained, if the trivial solution of the linear approximation system to solutions of a nonlinear system that is locally componentwise asymptotically equivalent to its linear approximation. There are given examples, that illustrate the application of proven sufficient conditions to the study of polystability of zero solutions of nonlinear systems of ordinary differential equations, including in the critical case, and also in the presence of positive eigenvalues.
Highlights
The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence
An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed
This operator satises the conditions of the Schauder principle, it has at least one xed point
Summary
Â. Âîñêðåñåíñêîãî î ïîêîìïîíåíòíîé àñèìïòîòè÷åñêîé ýêâèâàëåíòíîñòè â ïðåäïîëîæåíèè, ÷òî ìåæäó íà÷àëüíûìè òî÷êàìè èññëåäóåìûõ ñèñòåì óñòàíàâëèâàåòñÿ ñîîòâåòñòâèå òîëüêî â îêðåñòíîñòè íóëåâîãî ïîëîæåíèÿ ðàâíîâåñèÿ; ââåäåíî îïðåäåëåíèå ëîêàëüíîé ïîêîìïîíåíòíîé àñèìïòîòè÷åñêîé ýêâèâàëåíòíîñòè, ïîëó÷åíû äîñòàòî÷íûå óñëîâèÿ àñèìïòîòè÷åñêîé óñòîé÷èâîñòè è óñòîé÷èâîñòè íóëåâûõ ðåøåíèé íåëèíåéíûõ ñèñòåì îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé ñ âîçìóùåíèÿìè â ôîðìå âåêòîðíûõ ïîëèíîìîâ, â òîì ÷èñëå â êðèòè÷åñêîì ñëó÷àå.  ðàáîòå [17] ââåäåíû îïðåäåëåíèÿ ïîëèóñòîé÷èâîñòè è ïîëèóñòîé÷èâîñòè ïî îòíîøåíèþ ê ÷àñòè ïåðåìåííûõ, ïîëó÷åíû äîñòàòî÷íûå óñëîâèÿ ïîëèóñòîé÷èâîñòè íóëåâîãî ðåøåíèÿ íà îñíîâàíèè ìåòîäà ôóíêöèé Ëÿïóíîâà.  íàñòîÿùåé ðàáîòå ïðåäñòàâëåíû äîñòàòî÷íûå óñëîâèÿ ïîëèóñòîé÷èâîñòè ïî îòíîøåíèþ ê ÷àñòè ïåðåìåííûõ â ñìûñëå ðàáîòû [17] íóëåâîãî ðåøåíèÿ íåëèíåéíûõ ñèñòåì îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé ñ äîñòàòî÷íî ãëàäêîé ïðàâîé ÷àñòüþ
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