A survey of cubic systems. (English) Zbl 0747.34019

This is a highly valuable up-to-date account of recent progress in the qualitative theory of planar differential systems \(\dot x=P(x,y)\), \(\dot y=Q(x,y)\) with cubic polynomials \(P\), \(Q\). The extensive survey contains a wealth of results obtained during the last decade by authors from many countries, but mainly from China where this subject is remarkably cultivated and promoted. The following topics are treated: 1) Critical points and their index configuration. 2) Existence, nonexistence, uniqueness, and number of limit cycles. 3) Necessary and sufficient conditions for centres; fine foci, and bifurcating limit cycles. 4) Cubic systems with invariant algebraic curves. 5) Cubic systems whose positive semi-orbits are bounded. 6) Algebraic classification of cubic systems. Most of the results reported here are taken from the 83 references listed in the paper, but some of them are new.
Reviewer: J.Hainzl (Kassel)


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