On the Number of Limit Cycles of a Class of Near-Hamiltonian Systems with a Nilpotent Center

In this paper, we consider a class of near-Hamiltonian systems with a nilpotent center, and study the number of limit cycles including algebraic limit cycles. We prove that there are at most [Formula: see text] large amplitude limit cycles if the first-order Melnikov function is not zero identically, including an algebraic limit cycle. Moreover, it can have [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text] small limit cycles. We also provide two examples as applications of our main results.