In this thesis, we present an expanded account of Hilbert's 16th problem for dir=ltr Hilbert's 16th problem asks for the maximum number of limit cycles that a polynomial vector field, for a given degree, in the plane can have. Although the problem is more than 100 years old it is not even known whether a uniform upper bound, only depending on the degree of the vector field, might exist, even not when the degree is two. In the year 2000, S. Smale added the question to his list of problems for the 21st century, but restricting it to the dir=ltr type="#_x0000_t75" . In this thesis, we consider to be a polynomial of degree , with a fixed but arbitrary natural number. The related Liénard equation is of degree .