Integrability of Liénard systems with a weak saddle

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• 作者：Armengol Gasull ; Jaume Giné
• 关键词：Center problem ; Analytic integrability ; Weak saddle ; Liénard equation
• 刊名：Zeitschrift für angewandte Mathematik und Physik
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：68
• 期：1
• 全文大小：
• 刊物主题：Theoretical and Applied Mechanics; Mathematical Methods in Physics;
• 出版者：Springer International Publishing
• ISSN：1420-9039
• 卷排序：68

文摘

We characterize the local analytic integrability of weak saddles for complex Liénard systems, \(\dot{x}=y-F(x),\)\(\dot{y}= ax\), \( 0\ne a\in \mathbb {C}\), with F analytic at 0 and \(F(0)=F'(0)=0.\) We prove that they are locally integrable at the origin if and only if F(x) is an even function. This result implies the well-known characterization of the centers for real Liénard systems. Our proof is based on finding the obstructions for the existence of a formal integral at the complex saddle, by computing the so-called resonant saddle quantities.