### All Issues

Vol.10, 2020
Vol.9, 2019
Vol.8, 2018
Vol.7, 2017
Vol.6, 2016
Vol.5, 2015
Vol.4, 2014
Vol.3, 2013
Vol.2, 2012
Vol.1, 2011
 Volume 5, Number 1, 2015, Pages 141-145                                                                DOI：10.11948/2015012 The Hilbert number of a class of differential equations Jaume Llibre,Ammar Makhlouf Keywords:Periodic orbit, averaging theory, trigonometric polynomial, Hilbert number. Abstract: The notion of Hilbert number from polynomial differential systems in the plane of degree $n$ can be extended to the differential equations of the form $\dfrac{dr}{d\theta}=\dfrac{a(\theta)}{\displaystyle \sum_{j=0}^{n}a_{j}(\theta)r^{j}} \eqno(*)$ defined in the region of the cylinder $(\tt,r)\in \Ss^1\times \R$ where the denominator of $(*)$ does not vanish. Here $a, a_0, a_1, \ldots, a_n$ are analytic $2\pi$--periodic functions, and the Hilbert number $\HHH(n)$ is the supremum of the number of limit cycles that any differential equation $(*)$ on the cylinder of degree $n$ in the variable $r$ can have. We prove that $\HHH(n)= \infty$ for all $n\ge 1$. PDF      Download reader