On a Dulac function for the Kukles system

To estimate the number and location of limit cycles of Kukles systems in a strip of the phase plane xOy, we develop a method for constructing the Dulac function in the form of a polynomial in the phase variable y with coefficients depending on the second phase variable x. The suggested method is regular for h ₂(x) = 0. The proof of the fact that the constructed function is a Dulac function is reduced to finding a specific number of positive functions that are linear combinations of known functions of the single phase variable x with arbitrary constants. To construct the Dulac function, we use a solution of the corresponding linear programming problem. In addition, we show that the presented approach is efficient from the practical viewpoint and permits one to obtain a global exact estimate for the number of limit cycles of above-mentioned systems in some cases.