The Hilbert number of a class of differential equations

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}{dtheta}=dfrac{a(theta)}{displaystyle sum_{j=0}^{n}a_{j}(theta)r^{j}} eqno(*)] defined in the region of the cylinder $(tt,r)in Ss^1times R$ where the denominator of $(*)$ does not vanish. Here $a, a_0, a_1, ldots, a_n$ are analytic $2pi$--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 $nge 1$.