Bifurcation of Limit Cycles in Two Given Planar Polynomial Systems

Bifurcation of limit cycles in two given 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 perturbed planar polynomial systems. The study reveals that each of the two systems has 8 limit cycles. By using method of numerical simulation, the distributed orderliness of the 8 limit cycles is observed, and their nicety places are determined. The study also indicates that each of the 8 limit cycles passes the corresponding nicety point. The results presented here are helpful for further investigating the Hilbert's 16th problem.