Bifurcation of limit cycles for three planar polynomial systems is investigated by using both qualitative analysis and numerical exploration. The investigation is based on detection functions which are particularly effective for the planar polynomial systems. The study reveals that the perturbed Hamiltonian system (5) has only 3 limit cycles, whereas each of the two perturbed integrable non-Hamiltonian systems (6)–(7) has 4 limit cycles. By using method of numerical simulation, the distributed orderliness of these limit cycles is observed, and their nicety places are determined. The study also indicates that each of these limit cycles passes the corresponding nicety point.