Àííîòàöèÿ. Ðàññìàòðèâàåòñÿ çàäà÷à íàõîaeäåíèÿ ëîêàëüíîãî íåíóëåâîãî èíòåãðàëüíîãî ìíî-ãîîáðàçèÿ íåëèíåéíîé (n + m) -ìåðíîé ñèñòåìû îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíå-íèé, íå ðàçðåøåííûõ îòíîñèòåëüíî ïðîèçâîäíûõ. Ïðåäïîëàãàåòñÿ, ÷òî ó èçó÷àåìîé ñèñòåìû èìååòñÿ n -ìåðíîå òðèâèàëüíîå èíòåãðàëüíîå ìíîãîîáðàçèå ïðè âñåõ çíà÷åíèÿõ ïàðàìåòðà, à ñîîòâåòñòâóþùàÿ ëèíåéíàÿ ïîäñèñòåìà èìååò m -ïàðàìåòðè÷åñêîå ñåìåéñòâî ïåðèîäè÷åñêèõ ðåøåíèé. Ýòî îçíà÷àåò, â ÷àñòíîñòè, ÷òî ëèíåéíàÿ ñèñòåìà íå îáëàäàåò ñâîéñòâîì ýêñïîíåíöè-àëüíîé äèõîòîìèè. Äîïóñêàåòñÿ, ÷òî ìàòðèöà ëèíåéíîãî ïðèáëèaeåíèÿ ñèñòåìû ïðè íóëåâîì çíà÷åíèè ïàðàìåòðà ÿâëÿåòñÿ ôóíêöèåé îò íåçàâèñèìîé ïåðåìåííîé. Ïðîáëåìà ñóùåñòâî-âàíèÿ èíòåãðàëüíîãî ìíîãîîáðàçèÿ ñâîäèòñÿ ê ïðîáëåìå ðàçðåøèìîñòè îïåðàòîðíûõ óðàâ-íåíèé â ïðîñòðàíñòâå îãðàíè÷åííûõ Ëèïøèö-íåïðåðûâíûõ ïåðèîäè÷åñêèõ âåêòîð-ôóíêöèé. Äëÿ äîêàçàòåëüñòâà íàëè÷èÿ èíòåãðàëüíîãî ìíîãîîáðàçèÿ èñõîäíàÿ ñèñòåìà ïîäâåðãàåòñÿ ëè-íåàðèçàöèè, ê êîòîðîé ïðèìåíÿåòñÿ ìåòîä ïðåîáðàçóþùåé ìàòðèöû. Ìåòîä ïðåîáðàçóþùåé ìàòðèöû óäàåòñÿ ðàñïðîñòðàíèòü â òîì ÷èñëå è íà ñëó÷àé îòñóòñòâèÿ ëèíåéíûõ ïî ïàðàìåò-ðó ÷ëåíîâ îïåðàòîðíûõ óðàâíåíèé. Ïîëó÷åíû äîñòàòî÷íûå óñëîâèÿ ñóùåñòâîâàíèÿ â îêðåñò-íîñòè ñîñòîÿíèÿ ðàâíîâåñèÿ ñèñòåìû n -ìåðíîãî íåíóëåâîãî ïåðèîäè÷åñêîãî èíòåãðàëüíîãî ìíîãîîáðàçèÿ.Êëþ÷åâûå ñëîâà: ìåòîä ïðåîáðàçóþùåé ìàòðèöû, èíòåãðàëüíîå ìíîãîîáðàçèå, ñèñòåìà îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé, îïåðàòîðíîå óðàâíåíèå, óìåíüøåíèå ðàçìåð-íîñòè ôàçîâîãî ïðîñòðàíñòâà.
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