First integrals of local analytic differential systems

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• 作者：Jaume Llibre^a ; ^{jllibre@mat.uab.cat} ; Chara Pantazi^b ; ^{Chara.Pantazi@upc.edu} ; Sebastian Walcher^c ; ^{walcher@mathA.rwth.aachen.de}
• 关键词：primary ; 34C20 ; 34C45 ; 34M25
• 刊名：Bulletin des Sciences Mathematiques
• 出版年：2012
• 期刊代码：7_00074497
• 类别：mt
• 出版时间：April-May, 2012
• 卷：136
• 期：3
• 页码：342-359
• 文件大小：158 K

摘要

We investigate formal and analytic first integrals of local analytic ordinary differential equations near a stationary point. A natural approach is via the Poincar茅-Dulac normal forms: If there exists a formal first integral for a system in normal form then it is also a first integral for the semisimple part of the linearization, which may be seen as 鈥渃onserved鈥?by the normal form. We discuss the maximal setting in which all such first integrals are conserved, and show that all first integrals are conserved for certain classes of reversible systems. Moreover we investigate the case of linearization with zero eigenvalues, and we consider a three-dimensional generalization of the quadratic Dulac-Frommer center problem.