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

In this paper, we study the global dynamics for a class of continuous piecewise Z(2)-equivariant cubic Hamiltonian vector fields with nilpotent bicenters at (+/- 1, 0). We consider these polynomial vector fields with a challenging case where the bicenters (+/- 1, 0) come from the combination of two nilpotent cusps separated by y = 0. We call it a cusp-cusp type. We use the Poincare 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 Poincare disc.