## Eleven small limit cycles in a cubic vector field.(English)Zbl 0837.34042

The main result of this paper is described precisely by its title. However, the title may suggest to some that the paper is elementary. On the contrary, the centents of the paper reveal a deep analysis of the general problem of the appearance of limit cycles from a center or a weak focus of a planar polynomial system. Moreover, while much of the work in this area is done using computer algebra, the present author analyzes a very complicated bifurcation and achieves the current record for the number of small amplitude limit cycles that can appear in a cubic system by pure analysis with no use of computers. Thus, this important paper should be read not only for a proof of the title result, but for the insight it provides into the general bifurcation problem for polynomial systems.

### MSC:

 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 34C23 Bifurcation theory for ordinary differential equations
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