The limit cycles of medium amplitude in a family of the Kukles perturbations systems

Background. Searching of the limit cycles of polynomial systems goes back to the second part of the 16th problem of Gilbert which is still not solved completely. Searching of the limit cycles was carried out by various methods among which we will note finding of Lyapunov quantities and a method of averaging. The purpose of this work is check of a possibility of application of a method of averaging of first order to the system of Kukles of the fourth order for finding of medium amplitude limit cycles at polynomial perturbation. Methods. Salomon Rebollo-Perdomo and Claudio Vidal studied the quadratic Kukles differential systems and received the analytical equations which allow to find small amplitude and medium amplitude limit cycles at quadratic perturbation. We consider the system of Kukles of the fourth order and we apply a similar method to finding of medium amplitude limit cycles. As in this case it was not succeeded to receive the precise analytical equations, approximate methods were applied. Results. It was succeeded to show that the “approximate” method of averaging of first order allows to find the limit cycles of medium amplitude which arise from periodic trajectories of the center in the system of Kukles of the fourth order. Conclusions. Thus, it is proved that in the systems of Kukles of the fourth order and particular type the method of averaging of first order can be applied to finding of the medium amplitude limit cycles lying in a homoclinic loop.