Il’yashenko, Yu. S. [McFaden, Harold] Finiteness theorems for limit cycles. Transl. from the Russian by H. H. McFaden. (English) Zbl 0743.34036 Translations of Mathematical Monographs. 94. Providence, RI: American Mathematical Society (AMS). ix, 228 p. (1991). Since the discovery of a mistake in the famous paper of I. G. Petrovskiĭ and E. M. Landis [Mat. Sb., n. Ser. 37(79), 209–250 (1955; Zbl 0065.07202)] and two (1,3) distributions of limit cycles of quadratic systems independently by two Chinese mathematicians in 1979, many mathematicians had paid attention in the past years to the classical work of H. Dulac [Bull. Soc. Math. Fr. 51, 45–188 (1923; JFM 49.0304.01)] on the finiteness of limit cycles of any definite polynomial differential system (real). It was Il’yashenko who first gave a counter-example to a main assertion in Dulac’s long paper and proved that Dulac’s theorem is still true when the corner points of the infinite poly-cycle are all elementary. In the present very nice book Il’yashenko gives a detailed proof further, saying that Dulac’s conclusion is true even in the general case. The chief tools he uses are what he calls “functional cochain” and “super-exact sequence”. The viewpoint is high and the techniques are delicate and profound. Although it is impossible to introduce the details of this book here, we would like to suggest the reader to read it with care, if he has interest in solving Hilbert’s 16th problem.Moreover, from the works of some Chinese mathematicians in recent years concerning the existence and number of limit cycles in the neighbourhood of a generalized focus of certain cubic systems, it seems to us that to study the most general configuration of the generalized focus of an n- polynomial system, to prove the finiteness of the generalized focal quantities of such focus and to find the order of it is now an extremely urgent important problem. Maybe, a part of this problem has intimate relations with Il’yashenko’s work. Reviewer: Yanqian Ye (Nanjing) Cited in 7 ReviewsCited in 30 Documents MSC: 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations Keywords:functional cochain; super-exact sequence; generalized focus; limit cycles; quadratic systems; finiteness; polynomial differential system; Dulac’s theorem Citations:Zbl 0065.07202; JFM 49.0304.01 PDFBibTeX XMLCite \textit{Yu. S. Il'yashenko}, Finiteness theorems for limit cycles. Transl. from the Russian by H. H. McFaden. Providence, RI: American Mathematical Society (1991; Zbl 0743.34036)