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Singular points and limit cycles of planar polynomial vector fields. (English) Zbl 0953.34021

The authors apply the “gluing” method to construct planar polynomial systems of the form

x ' =P(x,y),y ' =Q(x,y),

with “large” numbers of limit cycles or singular points. For example, they show that there exists an absolute constant C with the following property: it is possible to construct a system with degP=degQ=d having at least

1 2log 2 d-Clog 2 log 2 d

limit cycles. They also show that, for any nonnegative integer numbers d,s 0 ,s 1 ,s 2 , and s 3 satisfying the conditions

2s 0 +s 1 +s 2 +s 3 =d 2 ,|s 1 +s 2 -s 3 |d,

it is possible to construct a system with degP=degQ=d having 2s 0 imaginary singular points, s 1 attractors, s 2 repellers, and s 3 saddles.

34C05Location of integral curves, singular points, limit cycles (ODE)
37C70Attractors and repellers, topological structure