Upper bounds for the number of limit cycles of some planar polynomial   differential systems
  We give an effective method for controlling the maximum number of limit cycles of some planar polynomial systems. It is based on a suitable choice of a Dulac function and the application of the well-known Bendixson-Dulac Criterion for multiple connected regions. The key point is a new approach to control the sign of the functions involved in the criterion. The method is applied to several examples. 
Armengol Gasull and Hector Giacomini