The cyclicity of period annulus of cubic isochronous Hamiltonian systems

Abstract

Cima, Ma\~{n}osas and Villadelprat (J. Differ. Equations, 157, 373--413, 1999) proved that a cubic Hamiltonian system possesses an isochronous center at the origin if and only if its Hamiltonian function can be expressed as \begin{eqnarray*}H_1(x,y)=k_1^2x^2+(k_2y+k_3x+k_4x^2)^2, \end{eqnarray*} where $k_1,k_2,k_3,k_4\in\mathbb{R}$, $k_1k_2\neq0$. This paper is devoted to investigating the weak Hilbert's 16th problem for the dynamical system associated with the above Hamiltonian function. We show that the maximum number of limit cycles is $n-1$. Furthermore, this number is reached. That is, we solve the weak Hilbert's 16th problem restricted to cubic Hamiltonian systems with an isochronous center at the origin.