Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

• Studying bifurcation of limit cycles related to Hilbert’s 16th problem.
• Showing the existence of 12 limit cycles around a singular point in a planar cubic system.
• This is the best result so far for planar cubic systems around a singular point.

In this paper, we prove the existence of 12 small-amplitude limit cycles around a singular point in a planar cubic-degree polynomial system. Based on two previously developed cubic systems in the literature, which have been proved to exhibit 11 small-amplitude limit cycles, we applied a different method to show 11 limit cycles. Moreover, we show that one of the systems can actually have 12 small-amplitude limit cycles around a singular point. This is the best result so far obtained in cubic planar vector fields around a singular point.