Phase Portraits of a Family of Hamiltonian Cubic Systems - Gouveia, Márcio R. A. [UNESP]; Llibre, Jaume; Roberto, Luci Any [UNESP] (Springer Nature, 2024)
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 (Formula presented.) where (x,y)∈R2 are the variables and μ is a real parameter. We classify in the Poincaré disc the topological phase portraits of this family of systems when the parameter μ varies, describing the bifurcations which take place.
Keywords: 37C29; 37D45; Bifurcations; Cubic polynomial differential systems; Cubic systems; Hamiltonian cubic systems; Poincaré compactification; Topological phase portraits