In this thesis based on paper \\cite{Yu?} , bifurcation of limit cycles is considered for planar cubic-order systems with an isolated nilpotent critical point . Normal form theory is applied to compute the generalized Lyapunov constants and to prove the existence of at least ? small-amplitude limit cycles in the neighborhood of the nilpotent critical point . In addition , the method of double bifurcation of nilpotent focus is used to show that such systems can have up to $??$ small-amplitude limit cycles near the nilpotent critical point . These are new lower bounds on the number of limit cycles in planar cubic-order systems near an isolated nilpotent critical point . Moreover , a set of center conditions is obtained for such cubic systems .
