center conditions and bifurcation of limit cycles at three-order nilpotent critical point in a septic lyapunov system

in this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of septic polynomial differential systems are investigated. with the help of computer algebra system mathematica, the first 13 quasi-lyapunov constants are deduced. as a result, sufficient and necessary conditions in order to have a center are obtained. the result that there exist 13 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 septic lyapunov systems.