Singular points and limit cycles of the generalized Kukles polynomial differential system

Background. Searching of numbers of Poincare limit cycles of polynomial dynamic systems belongs to second part of the 16$^{th}$ Gilbert problem, which is not solved in general. The purpose of this work is generalization of earlier results for the generalized Kukles system and new estimation of numbers of limit cycles the Kukles system 10 degree is got. Materials and methods. The methods of qualitative theory of dynamic systems and averaging theory were applied. Results. Singular points were researched of the generalized Kukles polynomial differential system and classification of phase portrait in the Poincare disc was showed. In addition, the program, which accelerated researching of numbers of limit cycles, was written using average theory. For the first time numbers of limit cycles for the Kukles system 10 degree depending on average degree are got. Conclusions. The classification of global phase portrait in the Poincare disc finishes a question about probable trajectory the generalized polynomial Kukles system. There is a potential for the future researching to get accurate assessment of numbers of limit cycles in respect to degree of the system without using of the program. In the future we are going to get analytic dependence numbers of limit cycles on system and average degrees.