How to estimate the number of limit cycles in Lienard systems with a small parameter

To estimate the number of limit cycles and locate them for polynomial Lienard systems with a small parameter in the case of a perturbation of a center and in the case of the existence of relaxation limit cycles, we develop a method for constructing a modified Dulac function in the form of a series in the small parameter. In the case of a perturbation of a center for Lienard systems, we suggest an heuristic method for the approximation of limit cycles appearing from the closed phase curves surrounding the center; in this method, we use the ovals obtained by equating the leading term in the expansion of the reciprocal of the integrating factor in powers of the small parameter with zero. The suggested method for finding Dulac functions permits one to single out one-parameter families of Lienard systems that have a constant (independent of the parameter) number of limit cycles. We present examples of such systems.