Limit cycles of differential equations

with P and Q polynomial and relatively prime, then the situation is substantially simpler, at least with respect to the richness of the asymptotic behaviour of the orbits. The Poincaré-Bendixson theorem implies that a bounded ω-limit set (and hence also a bounded α-limit set) of an orbit has to be either a singularity (also called a zero, an equilibrium, a critical point, or a stationary point), a periodic orbit, or a graphic, i.e., a connected set consisting of a finite number of singularities and a finite number of regular orbits that are homoclinic or heteroclinic between these singularities. Unbounded ωand α-limit sets can have a slightly more complicated structure but are still very simple. To get a fairly good idea of the phase portrait of a differential system (P,Q) as in (1) it suffices to draw the separatrix skeleton and extended separatrix skeleton, as precisely defined in [DLA]. The first essential ingredient of these skeletons is the set of singularities whose cardinality is bounded by mn (Bézout’s theorem), where m and n are the respective degrees of P and Q (we can suppose that m ≤ n). Besides the singularities the separatrix skeleton also contains a finite number of separatrices emanating from the singularities. These separatrices can be found by desingularization (also called blow-up) of the singularities. The computer program P4 is able to draw such separatrix skeletons. See [DLA] for an introduction to the use of P4 as well as to a detailed overview of the methods and theoretical results on which it is based. In extending the separatrix skeleton, the most difficult task consists of adding the periodic orbits. In 2002 a nice survey paper [I2] on this subject appeared in the Bulletin of the American Mathematical Society by the hand of Ilyashenko. I have to recall some information from that paper in order to precisely situate the contents of the book in review. I will however limit this to a minimum and will concentrate on supplementary results and more recent developments. I also do not want to go into detail on the subjects that are presented in the book. Periodic orbits in polynomial planar differential systems can be isolated or belong to an annulus of periodic orbits. The former are called limit cycles. If all orbits in the neighbourhood have the limit cycle C as ω-limit set (resp., α-limit set) we call C stable or attracting (resp., unstable or repelling). If all orbits on one side have C as ω-limit set and all orbits on the other side have C as α-limit set we call C semi-stable.