Phase Portraits of a Family of Hamiltonian Cubic Systems

While all the phase portraits of the quadratic polynomial Hamiltonian systems in the Poincaré disc were classified in 1994 (see Artés and Llibre (J Differ Equ 107: 80–95, 1994)), we are far from the classification of the phase portraits of the cubic polynomial Hamiltonian systems in the Poincaré disc. In this paper, we deal with the one-parameter family of cubic polynomial Hamiltonian systems \(\begin{aligned} {\dot{x}}=y-y(y^2+3x^2\mu ), \;\;\quad {\dot{y}}=x+x(x^2+3y^2\mu ), \end{aligned}\) $$\begin{array}{c}\hfill \dot{x}=y-y(y^2+3x^2\mu ),\dot{y}=x+x(x^2+3y^2\mu ),\end{array}$$ where \((x,y)\in \mathbb {R}^2\) $(x,y)\in {\mathbb{R}}^2$ are the variables and \(\mu\) $\mu$ is a real parameter. We classify in the Poincaré disc the topological phase portraits of this family of systems when the parameter \(\mu\) $\mu$ varies, describing the bifurcations which take place.