Abstract
The search for solutions of nonlinear stationary systems of ordinary differential equations (ODE) is sometimes very complicated. It is not always possible to obtain a general solution in an analytical form. As a consequence, a qualitative theory of nonlinear dynamical systems has been developed. Its methods allow us to investigate the properties of solutions without finding a general solution. Numerical methods of investigation are also widely used.In the case when it is impossible to find an analytically general solution of the ODE system, sometimes, nevertheless, it is possible to find its first integral. There is a number of known results that make it possible to obtain the first integral for certain special cases.The article deals with the method for obtaining the first integrals of ODE systems of the third order, based on the fact of integrability of the involutive distribution.The method proposed in the paper allows us to obtain the first integral of a nonlinear ODE system of the third order in the case when a vector field, which generates an involutive distribution of dimension 2 together with the vector field of the right-hand side of a given ODE system, is known. In this case, the solution of a certain sequence of Cauchy problems allows us to construct a level surface of the function of the first integral containing the given point of the state space of the system. Using the method of least squares, in a number of cases it is possible to obtain an analytic expression for the first integral.The article gives examples of the method application to two ODE systems, namely to a simple nonlinear third-order system and to the Lorentz system with special parameter values. The article shows how the first integrals can be obtained analytically using the method developed for the two systems mentioned above.
Highlights
Íåêîòîðûå ïðåäâàðèòåëüíûå ñâåäåíèÿÈçâåñòíî [7], ÷òî ñèñòåìà (1) â íåêîòîðîé îêðåñòíîñòè ëþáîé íåîñîáîé òî÷êè x0
Ñòàòüÿ ïîñâÿùåíà ÷èñëåííîìó ñïîñîáó ïîëó÷åíèÿ ïåðâûõ èíòåãðàëîâ ñèñòåì ÎÄÓ òðåòüåãî ïîðÿäêà, îñíîâàííîìó íà ôàêòå èíòåãðèðóåìîñòè èíâîëþòèâíîãî ðàñïðåäåëåíèÿ
The article shows how the first integrals can be obtained analytically using the method developed for the two systems mentioned above
Summary
Èçâåñòíî [7], ÷òî ñèñòåìà (1) â íåêîòîðîé îêðåñòíîñòè ëþáîé íåîñîáîé òî÷êè x0 Çàìåòèì, ÷òî â íåêîòîðîé îêðåñòíîñòè íåîñîáîé òî÷êè x0 ñóùåñòâóåò äèôôåîìîðôèçì y = T (x), ñïðÿìëÿþùèé âåêòîðíîå ïîëå F (x) [7]. Ò ò òîãäà ìîæíî, âûáðàâ H1(y) = (0, 1, 0) è H2(y) = (0, 0, 1) , ïîëó÷èòü äâà ëèíåéíî íåçàâèñèìûõ âåêòîðíûõ ïîëÿ, êîììóòèðóþùèõ ñî ñïðÿìëííûì âåêòîðíûì ïîëåì H(y) =. Ýòî îçíà÷àåò, ÷òî â íåêîòîðîé îêðåñòíîñòè òî÷êè T (x0) èõ ïîòîêè êîììóòèðóþò, ò.å. Î÷åâèäíî, ÷òî ïîòîêè ýòèõ âåêòîðíûõ ïîëåé â èñõîäíûõ ïåðåìåííûõ x òàêæå áóäóò êîììóòèðîâàòü; ýòî ïîêàçûâàåò, ÷òî ñóùåñòâóþò äâà ëèíåéíî íåçàâèñèìûõ âåêòîðíûõ ïîëÿ, êàæäîå èç êîòîðûõ. Èçâåñòíî, ÷òî â ñîîòâåòñòâèè ñ òåîðåìîé Ôðîáåíèóñà [9] äëÿ èíâîëþòèâíîé ñèñòåìû èç k ëèíåéíî íåçàâèñèìûõ âåêòîðíûõ ïîëåé ñóùåñòâóåò n − k ôóíêöèîíàëüíî íåçàâèñèìûõ ïåðâûõ èíòåãðàëîâ ýòîé ñèñòåìû. Ïîçâîëÿþùèé ïîñòðîèòü ïîâåðõíîñòè óðîâíÿ V (x) ñ ïîìîùüþ âû÷èñëåíèÿ êîìïîçèöèè ïîòîêîâ ñèñòåì (1) è (4), à â íåêîòîðûõ ñëó÷àÿõ ïîëó÷èòü àíàëèòè÷åñêîå âûðàæåíèå äëÿ ïåðâîãî èíòåãðàëà
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