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;...
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