Bifurcation of limit cycles in a cubic-order planar system around a nilpotent critical point

Abstract

In this paper, 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 9 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 10 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.