Some Methods for Complementing Weakly Regular Linear Extensions of Dynamical Systems to Regular Systems

Kulyk, V. L.; Pączko, D.

doi:10.1007/s10958-014-1702-9

Cited by 2 publications

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“…In many interesting investigations [1]- [4] it is arisen linear systems of differential equations in which we have to find the strong properties, i.e. such properties which arent changed under small perturbations.…”

Section: Introductionmentioning

confidence: 99%

Regularity of Linear Systems of Differential Equations on the Axes and Pencils of Quadratic Forms

Stepanenko¹

2019

Communications in Advanced Mathematical Sciences

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It is considered linear systems of differential equations and investigated questions of regularity of these systems. To explore the regularity it is comfortable to use quadratic form whose derivative with respect to the adjoint system is positive definite. Sometimes it is possible to find such a quadratic form, the derivative of which with respect to the system is non-negative. There are examples showing that in this case we can't say anything about the exponential dichotomy of this system (that is, its regularity). The question arises whether it is possible to combine a certain set of quadratic forms to get such a form, the derivative of which with respect to the system is positive definite. This question is similar to the question that arises in the theory of control: having a set of certain data about an object, can one say something about this object as a whole. It turns out that this is possible, only a set of these quadratic forms should be special, in some sense complete. In the presented article the authors propose to write it with the help of some combination of specific symmetric matrices S 1 , S 2 ,. .. . So we have a quadratic form V p = p 1 S 1 (t) x, x + p 2 S 2 (t) x, x + • • • + p k−1 S k−1 (t) x, x + S k (t) x, x It is proved that the derivative of this quadratic form is positive definite for sufficiently large parameters p 1 ,. .. , p k−1. The results are illustrated by examples.

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Section: Introductionmentioning

confidence: 99%

Regularity of Linear Systems of Differential Equations on the Axes and Pencils of Quadratic Forms

Stepanenko¹

2019

Communications in Advanced Mathematical Sciences

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“…Òàêi ôîðìè ìîaeóòü çìiíþâàòè çíàê i âèðîäaeóâàòèñü â äåÿêèõ òî÷êàõ, à¨õ ïîõiäíà â ñèëó ñèñòåìè ðiâíÿíü ¹ çíàêîâèçíà÷åíîþ. Çíàííÿ êîíêðåòíîãî âèãëÿäó ôóíêöi¨Ëÿïó-íîâà äîçâîëÿ¹ îöiíèòè âåëè÷èíó çáóðåííÿ äèíàìi÷íèõ ñèñòåì, ïðè ÿêèõ çáåðiãàþòüñÿ îáìåaeåíi iíâàðiàíòíi ìíîãîâèäè, à òîìó ÷àñòî ñòàâèòüñÿ çàäà÷à ïîáóäîâè òàêèõ ôóíêöiË ÿïóíîâà [4,5,9]. Íå äèâëÿ÷èñü íà ãëèáîêi òåîðåòè÷íi ðåçóëüòàòè, ÿêi îòðèìàíi â öüîìó íàïðÿìêó [7,8,9], ïèòàííÿ ïîáóäîâè ôóíêöi¨Ãðiíà i ôóíêöi¨Ëÿïóíîâà äëÿ ëiíiéíèõ ðîçøèðåíü äèíàìi÷íèõ ñèñòåì íà òîði çàëèøà¹òüñÿ äîñi àêòóàëüíèìè.…”

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Linearized Differential Equations Around a Multidimensional Tor

Grod¹,

Kulyk²,

Stepanenko³

2020

BMJ

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Àíàëiçó¹òüñÿ ìåòîä çàñòîñóâàííÿ êâàäðàòè÷íèõ ôîðì äî äîñëiäaeåííÿ ðåãóëÿðíîñòi ëiíiéíèõ ðîçøèðåíü äèíàìi÷íèõ ñèñòåì íà òîði. Çàïðîïîíîâàíî ìåòîä ïîáóäîâè ìíîaeèíè ðåãóëÿðíèõ ñèñòåì ç ôiêñîâàíîþ ôóíêöi¹þ Ëÿïóíîâà. Êëþ÷îâi ñëîâà i ôðàçè: Êëþ÷îâi ñëîâà i ôðàçè: Êâàäðàòè÷íà ôîðìà, ôóíêöiÿ Ãðiíà-Ñàìîéëåíêà, ðåãóëÿðíiñòü ñèñòåìè, ôóíêöiÿ Ëÿïóíîâà..

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Some Methods for Complementing Weakly Regular Linear Extensions of Dynamical Systems to Regular Systems

Cited by 2 publications

References 1 publication

Regularity of Linear Systems of Differential Equations on the Axes and Pencils of Quadratic Forms

Regularity of Linear Systems of Differential Equations on the Axes and Pencils of Quadratic Forms

Linearized Differential Equations Around a Multidimensional Tor

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