The center problem and bifurcations of limit cycles at a fifth-order nilpotent critical point in a Lyapunov system are studied. The concepts of focal values and Lyapunov constants as well as their computational method are given. As an example, a class of quintic systems that bifurcates at least 4 limit cycles enclosing the origin (node) is constructed.

center problem, Lyapunov constant, focal value, limit cycles, multiple Hopf bifurcation.