The center conditions and bifurcation of limit cycles at the degenerate singularity of a three-dimensional system

We investigate multiple limit cycles bifurcation and center-focus problem of the degenerate equilibrium for a three-dimensional system. by applying the method of symbolic computation,we obtain the first four quasi-lyapunov constants. it is proved that the system can generate 3 small limit cycles from nilpotent critical point on center manifold. furthermore,the center conditions are found and as weak foci the highest order is proved to be the fourth; thus we obtain at most 3 small limit cycles from the origin via local bifurcation. to our knowledge,it is the first example of multiple limit cycles bifurcating from a nilpotent singularity for the flow of a high-dimensional system restricted to the center manifold. © 2014 shugang song et al.