Two different distributions of limit cycles in a quintic system

In this paper, the conditions for bifurcations of limit cycles from a thirdorder nilpotent critical point in a class of quintic systems are investigated. Treaty the system coefficients as parameters, we obtain explicit expressions for the first fourteen quasi Lyapunov constants. As a result, fourteen or fifteen small amplitude limit cycles with different distributions could be created from the thirdorder nilpotent critical point by two different perturbations.