Document Zbl 0903.34002

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Math. Anal. Appl.</a> <a href="/?q=in%3A46131" title="Articles in this Issue">217, No. 2, 372-394 (1998)</a>. </div> <div class="abstract">In this clear and well-writen article the authors develop a Floquet theory for index-1 differential algebraic equations (DAEs). The theorem of Floquet on the representation of the fundamental matrix [<span class="zbmathjax-textit">G. Floquet</span>, Ann. Sci. Éc. Norm. Supér. 12, 47-89 (1883; <a href="/15.0279.01">JFM 15.0279.01</a>)] and the reduction theorem of <span class="zbmathjax-textit">A. M. Lyapunov</span> [The general problem of the stability of motion, London (1992; <a href="/0786.70001">Zbl 0786.70001</a>) (Originally: Kharkow, 1892) (Russian)], and, finally, the theorem on the stability of periodic solutions are generalized for DAEs. The authors show that the results for DAEs are as clear and simple as for regular ordinary differential equations (ODEs). <br class="zbmathjax-paragraph">In contrast to regular ODEs, DAEs have only implicitly given dynamic parts. To work out and utilize this implicit structure, a decoupling technique is applied, which goes back to <span class="zbmathjax-textit">E. Griepentrog</span> and <span class="zbmathjax-textit">R. März</span> [Differential-algebraic equations and their numerical treatment, Teubner-Texte Math. 88, Leipzig: BSB B. G. Teubner (1986; <a href="/0629.65080">Zbl 0629.65080</a>)]. Via a proper decoupling, the classical procedures of ODE theory [see for example <span class="zbmathjax-textit">L. S. Pontrjagin</span>, Gewöhnliche Differentialgleichungen, Mathematik für Naturwissenschaft und Technik. 11. Berlin: VEB Deutscher Verlag der Wissenschaften (1965; <a href="/0134.30201">Zbl 0134.30201</a>)] become applicable. The authors prove, that each linear DAE with periodic coefficients is periodically equivalent to a constant coefficient DAE in Kronecker normal form. The article contains a result on the stability of the trivial solution in the case of nonautonomous DAEs with a constant linear part and a small nonlinearity.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A7951">S.Siegmund (Augsburg)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1115476">Cited in <strong>17</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A09" title="MSC2020">34A09</a> </td> <td class="space"> Implicit ordinary differential equations, differential-algebraic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34C25" title="MSC2020">34C25</a> </td> <td class="space"> Periodic solutions to ordinary differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adifferential+algebraic+equations">differential algebraic equations</a>; <a href="/?q=ut%3AFloquet+theory">Floquet theory</a>; <a href="/?q=ut%3Amonodromy+matrix">monodromy matrix</a>; <a href="/?q=ut%3Astability+of+periodic+solutions">stability of periodic solutions</a>; <a href="/?q=ut%3AJFM+15.0279.01">JFM 15.0279.01</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0786.70001">Zbl 0786.70001</a>; <a href="/0629.65080">Zbl 0629.65080</a>; <a href="/0212.43002">Zbl 0212.43002</a>; <a href="/0134.30201">Zbl 0134.30201</a>; <a href="/15.0279.01">JFM 15.0279.01</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0903.34002" data-ciurl="/ci/01115476" data-biburl="/bibtex/01115476.bib" data-amsurl="/amsrefs/01115476.bib" data-xmlurl="/xml/01115476.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01115476.pdf" title="Zbl 0903.34002 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1006/jmaa.1997.5714" aria-label="DOI for “How Floquet theory applies to index-1 differential algebraic equations”" title="10.1006/jmaa.1997.5714">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Barz, B.; Suschke, E., Numerische Behandlung eines Algebro-Differentialgleichungssystems, RZ-Mitteilungen 7 (1994)</td> </tr><tr> <td>[2]</td> <td class="space">Hanke, M.; Griepentrog, E.; März, R., Berlin Seminar on Differential-Algebraic Equations, Seminarbericht 92-1 (1992) · <a href="/1541.34002" class="nowrap">Zbl 1541.34002</a></td> </tr><tr> <td>[3]</td> <td class="space">Farkas, M., Periodic Motions (1994), Springer: Springer New York · <a href="/0805.34037" class="nowrap">Zbl 0805.34037</a></td> </tr><tr> <td>[4]</td> <td class="space">Floquet, G., Sur les équations differentielles linéaires à coefficeints periodiques, Ann. Sci. École Norm. Sup., 12, 47-89 (1883) · <a href="/15.0279.01" class="nowrap">JFM 15.0279.01</a></td> </tr><tr> <td>[5]</td> <td class="space">Griepentrog, E.; März, R., Differential-Algebraic Equations and Their Numerical Treatment, Teubner-Texte Math. 88 (1986), Teubner: Teubner Leipzig · <a href="/0629.65080" class="nowrap">Zbl 0629.65080</a></td> </tr><tr> <td>[6]</td> <td class="space">Hale, J., Ordinary Differential Equations (1969), Wiley: Wiley New York · <a href="/0186.40901" class="nowrap">Zbl 0186.40901</a></td> </tr><tr> <td>[7]</td> <td class="space">R. Lamour, MSHDAE, www.mathematik.hu-berlin.de/serv/soft.html; R. Lamour, MSHDAE, www.mathematik.hu-berlin.de/serv/soft.html</td> </tr><tr> <td>[8]</td> <td class="space">Lyapunov, A. M., The General Problem of the Stability of Motion (1992), Taylor & Francis: Taylor & Francis London · <a href="/0786.70001" class="nowrap">Zbl 0786.70001</a></td> </tr><tr> <td>[9]</td> <td class="space">März, R., Numerical methods for differential-algebraic equations, Acta Numerica, 141-198 (1992) · <a href="/0769.65044" class="nowrap">Zbl 0769.65044</a></td> </tr><tr> <td>[10]</td> <td class="space">März, R., On linear differential-algebraic equations and linearizations, APNUM, 18, 267-292 (1995) · <a href="/0840.65071" class="nowrap">Zbl 0840.65071</a></td> </tr><tr> <td>[11]</td> <td class="space">Müller, P. C., Stabilität mechanischer Systeme mit holonomen Bindungen, Z. Angew. Math. Mech., 75, 593-594 (1995) · <a href="/0842.70013" class="nowrap">Zbl 0842.70013</a></td> </tr><tr> <td>[12]</td> <td class="space">Pontryagin, L. S., Gewöhnliche Differentialgleichungen (1965), Dt. Verl. Wiss: Dt. Verl. Wiss Berlin · <a href="/0134.30201" class="nowrap">Zbl 0134.30201</a></td> </tr><tr> <td>[13]</td> <td class="space">Mattheij, R. M.M.; Lamour, R.; März, R., On the stability behaviour of systems obtained by index-reduction, J. Comp. Appl. Math., 56, 305-319 (1994) · <a href="/0823.65075" class="nowrap">Zbl 0823.65075</a></td> </tr><tr> <td>[14]</td> <td class="space">Seydel, R., Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos (1994), Springer: Springer New York · <a href="/0806.34028" class="nowrap">Zbl 0806.34028</a></td> </tr><tr> <td>[15]</td> <td class="space">Tischendorf, C., On the stability of solutions of autonomous index-1 tractable and quasilinear index-2 tractable daes, Circuits Systems Signal Process, 13, 139-154 (1994) · <a href="/0801.34005" class="nowrap">Zbl 0801.34005</a></td> </tr><tr> <td>[16]</td> <td class="space">C. Tischendorf, Solution of Index-2 Differential Algebraic Equations and Its Application in Circuit Simulation, Humboldt University, Berlin, 1996; C. Tischendorf, Solution of Index-2 Differential Algebraic Equations and Its Application in Circuit Simulation, Humboldt University, Berlin, 1996</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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