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

Abstract

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 eightorder 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.