Local bifurcation of limit cycles and center problem for a class of quintic nilpotent systems

摘要

For a class of fifth degree nilpotent system, the shortened expressions of the first eight quasi-Lyapunov constants are presented. It is shown that the origin is a center if and only if the first eight quasi-Lyapunov constants are zeros. Under a small perturbation, the conclusion that eight limit cycles can be created from the eight-order weakened focus is vigorously proved. It is different from the usual Hopf bifurcation of limit cycles created from an elementary critical point. Mathematical Subject Classification: 34C07; 37G10.