Qualitative theory of planar differential systems

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.