Global phase portraits and bifurcation diagrams for Hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the [formula omitted]-axis

Abstract

This paper studies the global dynamics for Hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y-axis. By linear changes of variables, it can be written as four systems. One of them has been characterized in Llibre (2018). In this paper, the remaining three systems are investigated. Firstly, the normal forms of these systems are given by an algebraic classification of their infinite singular points. Then, we classify the global phase portraits of these systems having canonical forms on the Poincaré disk. Moreover, we obtain the bifurcation diagrams for the corresponding global phase portraits. Combining this work with Llibre (2018), we provide the complete classification of global phase portraits for Hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y-axis.