Bifurcation of limit cycles at a nilpotent critical point in a septic Lyapunov system

In this paper, we characterize local behavior of an isolated nilpotent critical point for a class of septic polynomial differential systems, including center conditions and bifurcation of limit cycles. With the help of computer algebra system-MATHEMATICA 12.0, the first 15 quasi-Lyapunov constants are deduced. As a result, necessary and sufficient conditions of such system having a center are obtained. We prove that there exist 16 small amplitude limit cycles created from the third-order nilpotent critical point. And then we give a lower bound of cyclicity of third-order nilpotent critical point for septic Lyapunov systems.