Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles

Abstract

<p style='text-indent:20px;'>We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian planar polynomial vector fields of degree 3 symmetric with respect to the <inline-formula><tex-math id="M1">\begin{document}$ x- $\end{document}</tex-math></inline-formula>axis having a nilpotent saddle at the origin.