Диссертация (Аналитико-численные методы исследования скрытых колебаний)

ÑÀÍÊÒ-ÏÅÒÅÐÁÓÐÃÑÊÈÉ ÃÎÑÓÄÀÐÑÒÂÅÍÍÛÉ ÓÍÈÂÅÐÑÈÒÅÒÍà ïðàâàõ ðóêîïèñèÊóçíåöîâ Íèêîëàé Âëàäèìèðîâè÷ÀÍÀËÈÒÈÊÎ-×ÈÑËÅÍÍÛÅ ÌÅÒÎÄÛÈÑÑËÅÄÎÂÀÍÈßÑÊÐÛÒÛÕ ÊÎËÅÁÀÍÈÉ05.13.18 ¾Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå, ÷èñëåííûå ìåòîäû è êîìïëåêñû ïðîãðàìì¿Äèññåðòàöèÿ íà ñîèñêàíèå ó÷¼íîé ñòåïåíèäîêòîðà ôèçèêî-ìàòåìàòè÷åñêèõ íàóêÍàó÷íûé êîíñóëüòàíò:äîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ÷ëåí-êîðð.

ÐÀÍËÅÎÍΠÃåííàäèé Àëåêñååâè÷Ñàíêò-Ïåòåðáóðã 20162ÎãëàâëåíèåÂâåäåíèå . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Ãëàâà 1. Ñêðûòûå àòòðàêòîðû . . . . . . . . . . . . . . . . . . . 121.1Àòòðàêòîðû äèíàìè÷åñêèõ ñèñòåì . . . . . . . . . . . . . . . . . .121.2Ñàìîâîçáóæäàþùèåñÿ è ñêðûòûå àòòðàêòîðû . . . . . . . .

. . .161.2.1Êëàññè÷åñêèå ïðèìåðû ñàìîâîçáóæäàþùèõñÿ àòòðàêòîðîâ171.2.1.1Êëàññè÷åñêàÿ ñèñòåìà Ëîðåíöà . . . . . . . . . . .171.2.1.2ÂîçìóùåííûéîñöèëëÿòîðÄóôôèíãà:ñàìîâîçáóæäàþùèéñÿ àòòðàêòîð Óåäà . . . . . . .1.2.2Ñêðûòûå àòòðàêòîðû â èçâåñòíûõ òåîðåòè÷åñêèõ ïðîáëåìàõ 201.2.2.11.2.2.21.2.319Ñêðûòûå êîëåáàíèÿ â 16-îé ïðîáëåìå ÃèëüáåðòàÑêðûòûåêîëåáàíèÿâêîíòðïðèìåðàõ20êãèïîòåçàì Àéçåðìàíà è Êàëìàíà . . . . . . . . . .20Ñêðûòûå àòòðàêòîðû â ïðèêëàäíûõ ìîäåëÿõ .

. . . . . . .241.2.3.1Ñêðûòûå àòòðàêòîðû â ýëåêòðîìåõàíè÷åñêèõñèñòåìàõ áåç ñîñòîÿíèÿ ðàâíîâåñèÿ . . . . . . . . .1.2.3.21.2.3.3Ñêðûòûå êîëåáàíèÿ ñ ìîäåëÿõ áóðîâûõ óñòàíîâîê 26Ñêðûòûå êîëåáàíèÿ â ñèñòåìàõ óïðàâëåíèÿëåòàòåëüíûìè àïïàðàòàìè . .

. . . . . . . . . . . .1.2.3.41.2.3.52429Ñêðûòûå àòòðàêòîðû â ñèñòåì ìîäåëè Ðàáèíîâè÷à 30Ñêðûòûé àòòðàêòîð â ìîäåëè ÃëóõîâñêîãîÄîëæàíñêîãî . . . . . . . . . . . . . . . . . . . . .341.2.3.6Ñêðûòûé àòòðàêòîð â ýëåêòðè÷åñêîé öåïè ×óà .341.2.3.7Ñêðûòûé àòòðàêòîð â ìîäåëè Ðàáèíîâè÷àÔàáðèêàíòà . . . . . . . . . . .

. . . . . . . . . . .3631.3Àíàëèòè÷åñêîå ïîñòðîåíèå ñêðûòûõ êîëåáàíèé â äâóìåðíûõïîëèíîìèàëüíûõ ñèñòåìàõ . . . . . . . . . . . . . . . . . . . . . . .381.3.0.8Îïðåäåëåíèå ëÿïóíîâñêèõ âåëè÷èí . . . . . . . . .381.3.0.9Ïðÿìîé ìåòîä âû÷èñëåíèÿ ëÿïóíîâñêèõ âåëè÷èí .461.3.0.10 Ìàëûå ïðåäåëüíûå öèêëû . . . . .

. . . . . . . . .501.4Àíàëèòèêî-÷èñëåííàÿ ïðîöåäóðà ëîêàëèçàöèè ñêðûòûõ êîëåáàíèé 531.5Ëîêàëèçàöèÿ ñêðûòûõ êîëåáàíèé â êîíòðïðèìåðàõ ê ïðîáëåìàìÀéçåðìàíà è Êàëìàíà . . . . . . . . . . . . . . . . . . . . . . . . .552 Ãëàâà 2. Ëÿïóíîâñêàÿ ðàçìåðíîñòü àòòðàêòîðîâ . . . . . . . . 772.1Ëÿïóíîâñêàÿ ðàçìåðíîñòü îòîáðàæåíèé è äèíàìè÷åñêèõ ñèñòåì .2.2Èíâàðèàíòíîñòüîòíîñèòåëüíîäèôôåîìîðôèçìîâèàíàëèòè÷åñêèå îöåíêè . . . . . . . . . . .

. . . . . . . . . . . . . .2.3Àíàëèòè÷åñêèåôîðìóëûòî÷íîéëÿïóíîâñêîé7884ðàçìåðíîñòèèçâåñòíûõ äèíàìè÷åñêèõ ñèñòåì . . . . . . . . . . . . . . . . . . .932.3.1Ñèñòåìà Ýíîíà . . . . . . . . . . . . . . . . . . . . . . . . .932.3.2Ñèñòåìà Ëîðåíöà . .

. . . . . . . . . . . . . . . . . . . . . .942.3.3Ñèñòåìà Ãëóõîâñêîãî-Äîëæàíñêîãî . . . . . . . . . . . . . .972.3.4Cèñòåìà ßíãà-Òèãàíà . . . . . . . . . . . . . . . . . . . . .992.3.5Cèñòåìà Øèìèöó-Ìîðèîêà . . . . . . . . . . . . . . . . . . 1003 Ãëàâà 3. Ñèñòåìû ôàçîâîé àâòîïîäñòðîéêè ÷àñòîòû: âûâîäìîäåëåé è ñêðûòûå êîëåáàíèÿ . . . . . .

. . . . . . . . . . . . . . 1023.1Âûâîä êëàññè÷åñêèõ ìîäåëåé ÔÀÏ â ïðîñòðàíñòâå ôàç ñèãíàëîâ . 1033.2Ïîíÿòèÿ äèàïàçîíîâ óñòîé÷èâîñòè . . . . . . . . . . . . . . . . . . 1113.2.1Ëîêàëüíàÿ óñòîé÷èâîñòü è ïîëîñà óäåðæàíèÿ . . . . . . . . 1143.2.2Ãëîáàëüíàÿ óñòîé÷èâîñòü è ïîëîñà çàõâàòà . . . . . . . .

. 1203.2.3Ïðîñêàëüçûâàíèåöèêëîâèïîëîñàçàõâàòàáåçïðîñêàëüçûâàíèÿ . . . . . . . . . . . . . . . . . . . . . . . . 1243.2.4Àïïðîêñèìàöèÿ ïîëîñû çàõâàòà áåç ïðîñêàëüçûâàíèÿ äëÿêëàññè÷åñêîé ÔÀÏ . . . . . . . . . . . . . . . . . . . . . . . 1283.2.5Íà÷àëüíàÿ è ñîáñòâåííàÿ ÷àñòîòû Ïà . . . . . .

. . . . . . 1323.3Ñêðûòûå êîëåáàíèÿ â äâóõôàçíîé ÔÀÏ . . . . . . . . . . . . . . . 1343.4Âûâîä ìàòåìàòè÷åñêèõ ìîäåëåé êëàññè÷åñêîé ñõåìû Êîñòàñà . . . 1424Çàêëþ÷åíèå . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Ñïèñîê ëèòåðàòóðû . . . . . . . . . . . . . . .

. . . . . . . . . . . . . 1525ÂâåäåíèåÊîëåáàíèå â äèíàìè÷åñêîé ñèñòåìå ìîæåò áûòü ëåãêî âû÷èñëåíî,åñëè òðàåêòîðèè ñ íà÷àëüíûìè äàííûìè èç åãî îòêðûòîé îêðåñòíîñòè âôàçîâîì ïðîñòðàíñòâå ïðèòÿãèâàþòñÿ ê ðàññìàòðèâàåìîìó êîëåáàíèþ ïðèâîçðàñòàíèè âðåìåíè.Ñ âû÷èñëèòåëüíîé òî÷êè çðåíèÿ òàêîå êîëåáàíèå(èëè ìíîæåñòâî òàêèõ êîëåáàíèé) íàçûâàåòñÿ àòòðàêòîðîì,åãî ïðèòÿæåíèÿ áàññåéíîì ïðèòÿæåíèÿ.à îáëàñòüÄëÿ âèçóàëèçàöèè òàêîãîàòòðàêòîðà äîñòàòî÷íî âûáðàòü íà÷àëüíóþ òî÷êó è ÷èñëåííî íàáëþäàòüçà ïåðåõîäíûì ïðîöåññîì ïðèòÿæåíèÿ òðàåêòîðèè èç âûáðàííîé òî÷êè êàòòðàêòîðó ñ âîçðàñòàíèåì âðåìåíè.

Èçó÷åíèå äèíàìè÷åñêèõ ñèñòåì îáû÷íîíà÷èíàåòñÿ ñ àíàëèçà ñîñòîÿíèé ðàâíîâåñèÿ, êîòîðûå ìîãóò áûòü ëåãêîíàéäåíû ÷èñëåííî èëè àíàëèòè÷åñêè.Ïîýòîìó ñ âû÷èñëèòåëüíîé òî÷êèçðåíèÿ åñòåñòâåííî ïðåäëîæèòü ñëåäóþùóþ êëàññèôèêàöèþ àòòðàêòîðîâ,êîòîðàÿ îñíîâàíà íà ñâÿçè èõ îáëàñòåé ïðèòÿæåíèÿ è ñîñòîÿíèé ðàâíîâåñèÿ:àòòðàêòîð íàçûâàåòñÿ ñàìîâîçáóæäàþùèìñÿ (self-excited attractor), åñëèëþáàÿ îêðåñòíîñòü îäíîãî èç ñîñòîÿíèé ðàâíîâåñèÿ ïåðåñåêàåòñÿ ñ îáëàñòüþïðèòÿæåíèÿ àòòðàêòîðà, â ïðîòèâíîì ñëó÷àå àòòðàêòîð íàçûâàåòñÿñêðûòûì (hidden attractor). Òåðìèíû self-excited attractor è hidden attractor áûëè ââåäåíû àâòîðîì â ðàáîòàõ [44, 160, 169, 180, 188] è çà ïîñëåäíèåãîäû ïîëó÷èëè øèðîêîå ðàñïðîñòðàíåíèå. Ñòàòüÿ [191] î ëîêàëèçàöèè ñêðûòîãîàòòðàêòîðà â öåïè ×óà, îïóáëèêîâàííàÿ â æóðíàëå Physics Letters A â 2011 ãîäó,ñòàëà â 2016 ãîäó ñàìîé öèòèðóåìîé ñòàòüåé æóðíàëà çà ïÿòü ëåò1 , îáçîðíàÿñòàòüÿ [180] 2013 ãîäà, ïîäãîòîâëåííàÿ ïî ìàòåðèàëàì äàííîé äèññåðòàöèîííîéðàáîòû äëÿ æóðíàëà International Journal of Bifurcation and Chaos in AppliedSciences and Engineering çà ïðîøåäøåå âðåìÿ ñòàëà ñàìîé ÷èòàåìîé ñòàòåé1 ñîãëàñíîèíôîðìàöèè íà îôèöèàëüíîì ñàéòå æóðíàëà íà 09.05.2016:http://www.journals.elsevier.com/physics-letters-a/most-cited-articles6æóðíàëà è âîøëà â ñïèñîê ñàìûõ öèòèðóåìûõ ðàáîò æóðíàëà çà âñå ãîäû2 ,à â 2016 ãîäó îáçîðíàÿ ñòàòüÿ Hiddenattractors in Dynamical systems[129]áûëà ïðèíÿòà â îäèí èç âåäóùèõ3 ìåæäóíàðîäíûõ æóðíàëîâ Physics Reports.Òàêàÿ êëàññèôèêàöèÿ àòòðàêòîðîâ îêàçàëàñü åñòåñòâåííîé íå òîëüêîäëÿ èññëåäîâàíèÿ òàêèõ ôóíäàìåíòàëüíûõ ïðîáëåì, êàê 16-àÿ ïðîáëåìàÃèëüáåðòà è ãèïîòåçû Àéçåðìàíà è Êàëìàíà, íî è îòðàçèëà òðóäíîñòèèíæåíåðíîãî àíàëèçà ðàçëè÷íûõ ìóëüòèóñòîé÷èâûõ ïðèêëàäíûõ íåëèíåéíûõìîäåëåé (â êîìïüþòåðíûõ àðõèòåêòóðàõ è òåëåêîììóíèêàöèè, ñèñòåìàõóïðàâëåíèÿ ëåòàòåëüíûìè àïïàðàòàìè,ýëåêòðîìåõàíè÷åñêèõ ìîäåëÿõ èò.ä.).

Ïðåäëîæåííàÿ êëàññèôèêàöèÿ ÿâèëàñü êàòàëèçàòîðîì îòêðûòèÿ íîâûõñêðûòûõ àòòðàêòîðîâ â ðàçëè÷íûõ ñèñòåìàõ.  íàñòîÿùåå âðåìÿ ïî òåìàòèêåñêðûòûõ àòòðàêòîðîâ îïóáëèêîâàíî áîëåå 100 ðàáîò èçâåñòíûõ ðîññèéñêèé èçàðóáåæíûõ àâòîðîâ, ñðåäè êîòîðûõ Erik Mosekilde (Äàíèÿ), Luigi Fortuna(Èòàëèÿ), Tomasz Kapitaniak (Ïîëüøà), Ivan Zelinka (×åõèÿ), Christos Volos (Ãðåöèÿ), William Heath (Âåëèêîáðèòàíèÿ), Manuel de la Sen (Èñïàíèÿ),Awadhesh Prasad (Èíäèÿ), Guanrong Chen (Êèòàé), Viet Pham (Âüåòíàì), Sajad Jafari (Iran), Ihsan Pehlivan (Òóðöèÿ), Sifeu Kingni (Êàìåðóí), Julien Clinton Sprott (ÑØÀ), Æ.Ò. Æóñóáàëèåâ (Êóðñê), Â.Ñ.

Àíèùåíêî (Ñàðàòîâ),È.Ì. Áóðêèí (Òóëà), À.Í. ×óðèëîâ (Ñàíêò-Ïåòåðáóðã), Ñ.Ï. Êóçíåöîâè À.Ï. Êóçíåöîâ (Ñàðàòîâ) è äðóãèõ, ãäå ïðåäëîæåííàÿ êëàññèôèêàöèÿàòçòðàêòîðîâ èñïîëüçóþòñÿ ñî ññûëêàìè íà ðàáîòû â ýòîì íàïðàâëåíèè àâòîðàäèññåðòàöèè [44,169,180,191,192]; â òîì ÷èñëå â áèáëèîìåòðè÷åñêîé áàçå äàííûõScopus (Elsevier) ïðîèíäåêñèðîâàíî áîëåå 40 ðàáîò çàðóáåæíûõ àâòîðîâ, âíàçâàíèÿõ êîòîðûõ ñîäåðæèòñÿ hidden attractor.  2015 ãîäó áûë îïóáëèêîâàíñïåöèàëüíûé âûïóñê æóðíàëà The European Physical Journal Special TopicsMultistability: Uncovering Hidden Attractors (Springer)4 , â êîòîðîì ïðèíÿëèó÷àñòèå àâòîðû èç 14 ñòðàí.  2015 ãîäó íà ìåæäóíàðîäíîé êîíôåðåíöèè 4thIFAC Conference on Analysis and Control of Chaotic Systems (Japan, 2015) Guan2 ñîãëàñíîèíôîðìàöèè íà îôèöèàëüíîì ñàéòå æóðíàëà íà 09.05.2016:http://www.worldscientific.com/worldscinet/ijbc?null&journalTabs=read ,http://www.worldscientific.com/worldscinet/ijbc?null&journalTabs=cited3 Impact factor 20.033 (JCR 2015, Web of Science), http://www.journals.elsevier.com/physics-reports/4 http://link.springer.com/journal/11734/topicalCollection/AC_9fcd6cfa51cbf272d9e9981e7d184d22/7rong Chen5 âûñòóïèë ñ ïëåíàðíûì äîêëàäîì, ïîñâÿùåííîì òåìàòèêå ñêðûòûõàòòðàêòîðîâ: attractors 6.ÒåðìèíChaotic systems with any number of equilibria and their hidden ñàìîâîçáóæäàþùèåñÿêîëåáàíèÿ(self-excitedoscillation)èñïîëüçîâàëñÿ â íàó÷íîé øêîëå À.À.

Àíäðîíîâà äëÿ îïèñàíèÿ ïåðåõîäíûõïðîöåññîâ, âîçíèêàþùèõ ïðè äîáàâëåíèè â àâòîíîìíóþ ñèñòåìó âíåøíåãîâîçìóùåíèÿ è ïåðåâîäÿùèõ ñîñòîÿíèå ñèñòåìû èç ñîñòîÿíèÿ ðàâíîâåñèÿ(àâòîíîìíîé ñèñòåìû) ê êîëåáàòåëüíîìó ðåæèìó (â íåàâòîíîìíîé ñèñòåìå).Èñïîëüçîâàíèå ýòîãî òåðìèíà â äàííîé ðàáîòå äëÿ àâòîíîìíûõ ñèñòåìõàðàêòåðèçóåò íàëè÷èå â ðàññìàòðèâàåìîé àâòîíîìíîé ñèñòåìå àíàëîãè÷íîãîåñòåñòâåííîãî ïåðåõîäíîãî ïðîöåññà èç îêðåñòíîñòè îäíîãî èç íåóñòîé÷èâûõñîñòîÿíèé ðàâíîâåñèÿ ê àòòðàêòîðó.Òåðìèí ñêðûòûåêîëåáàíèÿ(hiddenoscillation) ïîÿâèëñÿ ïðè îáñóæäåíèè [159, 166, 185] âîçìîæíîñòè âèçóàëèçàöèèïðåäåëüíûõ öèêëîâ â ïîëèíîìèàëüíûõ ñèñòåìàõ äëÿ èññëåäîâàíèÿ 16-îéïðîáëåìû Ãèëüáåðòà.

 ïåðâûõ èçâåñòíûõ ïðèìåðàõ äâóìåðíûõ êâàäðàòè÷íûõñèñòåì ñ ÷åòûðüìÿ ïðåäåëüíûìè öèêëàìè, êîòîðûå ñòàëè êîíòðïðèìåðàìè êóòâåðæäåíèþ È.Ã. Ïåòðîâñêîãî è Å.Ì. Ëàíäèñ î âîçìîæíîñòè ñóùåñòâîâàíèÿòîëüêî òðåõ ïðåäåëüíûõ öèêëîâ â òàêèõ ñèñòåìàõ, ÷èñëåííî ìîæíî óâèäåòüòîëüêî îäèí ïðåäåëüíûé öèêë (íîðìàëüíîé àìïëèòóäû), òîãäà êàê òðèîñòàëüíûõ öèêëà (ìàëîé àìïëèòóäû) ñòðîÿòñÿ òåîðåòè÷åñêè ïðè ïîìîùèïîñëåäîâàòåëüíî ìàëûõ âîçìóùåíèé è, ïîýòîìó, íå ìîãóò áûòü âèçóàëèçîâàíûñòàíäàðòíûìè ñðåäñòâàìè7 .Äðóãèì ìîòèâîì èñïîëüçîâàíèÿ òåðìèíà ñêðûòûå ñòàëî îïèñàíèå Â.È.

Àðíîëüäîì8 ýêñïåðèìåíòà À.Í. Êîëìîãîðîâà,êîòîðûé ðàçäàâàë ñòóäåíòàì ÌÃÓ â êà÷åñòâå ïðàêòèêóìà äâóìåðíûåêâàäðàòè÷íûå ñèñòåìû ñ ðàçëè÷íûìè êîýôôèöèåíòàìè äëÿ èññëåäîâàíèÿñóùåñòâîâàíèÿ â íèõ ïðåäåëüíûõ öèêëîâ â íåñêîëüêèõ ñîòíÿõ òàêèõïðèìåðîâ ñòóäåíòû íå íàøëè íè îäíîãî ïðåäåëüíîãî öèêëà. Ìàòåìàòè÷åñêàÿôîðìàëèçàöèÿýòîãîïîíÿòèÿáûëàñâÿçàíàñïðèìåðîìïîñòðîåíèÿõàîòè÷åñêîãî ñêðûòîãî àòòðàêòîðà (hidden attractor) â ýëåêòðîííîé öåïè5 CityUniversity of Hong-Kong, China, http://www.ee.cityu.edu.hk/~gchen/ ; IEEE Fellow,Highly Cited Researchers 2015 in Engineering and Mathematics (Thomson Reuters).6 http://www.ee.cityu.edu.hk/~gchen/pdf/CHEN_IFAC2015.pdf7 S.

Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica, 23.1980, 153-158.8 Â.È. Àðíîëüä, Ýêñïåðèìåíòàëüíàÿ ìàòåìàòèêà. Ì.: Ôàçèñ. 2005.8×óà (Chua circuit) ñ èñêóññòâåííî ñòàáèëèçèðîâàííûì íóëåâûì ñîñòîÿíèåìðàâíîâåñèÿ, êîòîðûé áûë ïîñòðîåí àâòîðîì è ïðåäñòàâëåí íà ïëåíàðíîìäîêëàäå 4th International Scientic Conference on Physics and Control â 2009ãîäó9 è çàòåì îïóáëèêîâàí â [169].Öåëè ðàáîòû.àòòðàêòîðîâ.ëîêàëèçàöèèÑîçäàíèå êîíöåïöèè ñêðûòûõ è ñàìîâîçáóæäàþùèõñÿÐàçâèòèåñêðûòûõýôôåêòèâíûõàòòðàêòîðîââàíàëèòèêî-÷èñëåííûõôóíäàìåíòàëüíûõìåòîäîâïðîáëåìàõèôèçè÷åñêèõ ìîäåëÿõ.