Coexistence of Algebraic Limit Cycles and Small Limit Cycles of Two Classes of Near-Hamiltonian Systems with a Nilpotent Singular Point

In this paper, two classes of near-Hamiltonian systems with a nilpotent center are considered: the coexistence of algebraic limit cycles and small limit cycles. For the first class of systems, there exist 2n+1 limit cycles, which include an algebraic limit cycle and 2n small limit cycles. For the second class of systems, there exist n2+3n+22 limit cycles, including an algebraic limit cycle and n2+3n2 small limit cycles.