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In this paper, we present the zero bifurcation diagrams for the Abelian integrals of two hyperelliptic Hamiltonian systems with three perturbation parameters using an algebraic-geometric approach. The method can be used to study the bifurcation diagrams for higher-order Hamiltonian systems with polynomial perturbations of any degree.

ZERO BIFURCATION DIAGRAMS FOR ABELIAN INTEGRALS: A STUDY ON HIGHER-ORDER HYPERELLIPTIC HAMILTONIAN SYSTEMS WITH THREE PERTURBATION PARAMETERS