The 16th Hilbert problem restricted to circular algebraic limit cycles

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• 作者：Jaume Llibre^a ; ^{jllibre@mat.uab.cat" class="auth_mail" title="E-mail the corresponding author} ; Rafael Ramí ; rez^b ; ^{rafaelorlando.ramirez@urv.cat" class="auth_mail" title="E-mail the corresponding author} ; Valentí ; n Ramí ; rez^c ; ^{vramirsa8@alumnes.ub.edu" class="auth_mail" title="E-mail the corresponding author} ; Natalia Sadovskaia^d ; ^{natalia.sadovskaia@upc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：34C07
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 April 2016
• 年：2016
• 卷：260
• 期：7
• 页码：5726-5760
• 全文大小：384 K

文摘

We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S   admits at most c20aa8fa28927928c14c1137c7e" title="Click to view the MathML source">S−1$S-1$ invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S   has at most c20aa8fa28927928c14c1137c7e" title="Click to view the MathML source">S−1$S-1$ algebraic limit cycles given by circles, and this number is reached.