Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations

* Correspondence: jinyinlai@lyu. edu.cn School of Science, Linyi University, Linyi 276005, Shandong, P.R. China Full list of author information is available at the end of the article Abstract In this article, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of seventh-degree 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 seventh-degree nilpotent systems. MSC: 34C05; 34C07.