On the Abel differential equations of third kind

Abstract

Abel equations of the first and second kind have been widely studied, but one question that never has been addressed for the Abel polynomial differential systems is to understand the behavior of its solutions (without knowing explicitly them), or in other words, to obtain its qualitative behavior. This is a very hard task that grows exponentially as the number of parameters in the equation increases. In this paper, using Poincare compactification we classify the topological phase portraits of a special kind of quadratic differential system, the Abel quadratic equations of third kind. We also describe the maximal number of polynomial solutions that Abel polynomial differential equations can have.