Nilpotent Bicenters in Continuous Piecewise ℤ2-Equivariant Cubic Polynomial Hamiltonian Vector Fields: Cusp–Cusp Type

Abstract

In this paper, we study the global dynamics for a class of continuous piecewise [Formula: see text]-equivariant cubic Hamiltonian vector fields with nilpotent bicenters at [Formula: see text]. We consider these polynomial vector fields with a challenging case where the bicenters [Formula: see text] come from the combination of two nilpotent cusps separated by [Formula: see text]. We call it a cusp–cusp type. We use the Poincaré compactification, the blow-up theory, the index theory and the theory of discriminant sequence for determining the number of distinct or negative real roots of a polynomial, to classify the global phase portraits of these vector fields in the Poincaré disc.