Center conditions and bifurcation of limit cycles at three-order nilpotent critical point in a cubic Lyapunov system

In the present paper, for the three-order nilpotent critical point of a cubic Lyapunov system, the center problem and bifurcation of limit cycles are investigated. With the help of computer algebra system-MATHEMATICA, the first 7 quasi-Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact of there exist 7 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth we give a lower bound of cyclicity of three-order nilpotent critical point for cubic Lyapunov systems.