Our aim is to study ordinary di?erential equations or simply di?erential s- tems in two real
variables x ? = P(x y) (0.1) y? = Q(x y) r 2 where P and Q are C functions de?ned on an open
subset U of R with ? r=1 2 ... ? ?.AsusualC standsforanalyticity.Weputspecialemphasis onto
polynomial di?erential systems i.e. on systems (0.1) where P and Q are polynomials. Instead
of talking about the di?erential system (0.1) we frequently talk about its associated vector
?eld ? ? X = P(x y) +Q(x y) (0.2) ?x ?y 2 on U? R . This will enable a coordinate-free approach
which is typical in thetheoryofdynamicalsystems.Anotherwayexpressingthevector?eldisby
writingitasX=(P Q).Infact wedonotdistinguishbetweenthedi?erential system (0.1) and its vector
?eld (0.2). Almost all the notions and results that we present for two-dimensional di?erential
systems can be generalized to higher dimensions and manifolds but our goal is not to present
them in general we want to develop all these notions and results in dimension 2. We would like
this book to be a nice introduction to the qualitative theory of di?erential equations in the
plane providing simultaneously the major part of concepts and ideas for developing a similar
theory on more general surfaces and in higher dimensions. Except in very limited cases we do
not deal with bifurcations but focus on the study of individual systems.
