Abstract
A functional method of localization has proved to be good in solving the qualitative analysis problems of dynamic systems. Proposed in the 90s, it was intensively used when studying a number of well-known systems of differential equations, both of autonomous and of non-autonomous discrete systems, including systems that involve control and / or disturbances.The method essence is to construct a set containing all invariant compact sets in the phase space of a dynamical system. A concept of the invariant compact set includes equilibrium positions, limit cycles, attractors, repellers, and other structures in the phase space of a system that play an important role in describing the behavior of a dynamical system. The constructed set is called localizing and represents an external assessment of the appropriate structures in the phase space.Relatively recently, it was found that the functional localization method allows one to analyze a behavior of the dynamical system trajectories. In particular, the localization method can be used to check the stability of the equilibrium positions.Here naturally emerges an issue of the relationship between the functional localization method and the well-known La Salle invariance principle, which can be regarded as a further development of the method of Lyapunov functions for establishing stability. The article discusses this issue.
Highlights
Ëþáàÿ òðàåêòîðèÿ xÍåïóñòîå êîìïàêòíîå ω-ïðåäåëüíîå ìíîæåñòâî ω(γ), ê êîòîðîìó îíà ñòðåìèòñÿ ïðè t → +∞ â òîïîëîãèè êîìïàêòíîãî ïðîñòðàíñòâà n
 ñòàòüå ïðîâîäèòñÿ ñðàâíåíèå äâóõ ïîäõîäîâ ê êà÷åñòâåííîìó àíàëèçó àâòîíîìíîé ñèñòåìû îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé: èñïîëüçîâàíèå ïðèíöèïà èíâàðèàíòíîñòè ËàÑàëëÿ è ôóíêöèîíàëüíûé ìåòîä ëîêàëèçàöèè èíâàðèàíòíûõ êîìïàêòîâ
A functional method of localization has proved to be good in solving the qualitative analysis problems of dynamic systems
Summary
Íåïóñòîå êîìïàêòíîå ω-ïðåäåëüíîå ìíîæåñòâî ω(γ), ê êîòîðîìó îíà ñòðåìèòñÿ ïðè t → +∞ â òîïîëîãèè êîìïàêòíîãî ïðîñòðàíñòâà n. Òîãäà ìîæíî óòâåðæäàòü, ÷òî äëÿ ëþáîé îãðàíè÷åííîé ñïðàâà (ïðè t → +∞) òðàåêòîðèè γ åå ω-ïðåäåëüíîå ìíîæåñòâî, êàê èíâàðèàíòíûé êîìïàêò, ñîäåðæèòñÿ â ìíîæåñòâå Ω(V, Q). Òàêæå îòìåòèì, ÷òî â äàííîì ïðèìåðå ïîëóïëîñêîñòü x ≥ 0 | èíâàðèàíòíîå ìíîæåñòâî è ôóíêöèÿ φ (x.y) íà ýòîì ìíîæåñòâå ïîëîæèòåëüíî ïîëóîïðåäåëåíà. Îòìåòèì, ÷òî ôóíêöèîíàëüíûé ìåòîä ëîêàëèçàöèè íå ìîæåò áûòü îáîñíîâàí ñ ïîìîùüþ ïðèíöèïà èíâàðèàíòíîñòè, ïîñêîëüêó â ýòîì ïðèíöèïå ðå÷ü èäåò î ïîâåäåíèè òðàåêòîðèé ïðè íåîãðàíè÷åííîì âîçðàñòàíèè âðåìåíè, â òî âðåìÿ êàê ôóíêöèîíàëüíûé ìåòîä ëîêàëèçàöèè äàåò îöåíêó èíâàðèàíòíûõ êîìïàêòíûõ ìíîæåñòâ, êîòîðûå ìîãóò áûòü è íå ñâÿçàíû ñ ïðåäåëüíûì ïåðåõîäîì ïî âðåìåíè.  ñòàòüå ïîêàçàíî, ÷òî äàæå â óñëîâèÿõ ïðèíöèïà èíâàðèàíòíîñòè (çíàêîâàÿ ïîëóîïðåäåëåííîñòü ïðîèçâîäíîé ôóíêöèè) ïðè äîïîëíèòåëüíîì óñëîâèè, ÷òî ôóíêöèÿ íà óíèâåðñàëüíîì ñå÷åíèè ïîñòîÿííà, óíèâåðñàëüíîå ñå÷åíèå è ëîêàëèçèðóþùåå ìíîæåñòâî ìîãóò íå ñîâïàäàòü. Ðàáîòà âûïîëíåíà ïðè ôèíàíñîâîé ïîääåðæêå Ìèíèñòåðñòâà íàóêè è âûñøåãî îáðàçîâàíèÿ (ïðîåêò 0705-2020-0047) è Ðîññèéñêîãî ôîíäà ôóíäàìåíòàëüíûõ èññëåäîâàíèé (20-0700294)
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