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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 $$ x'=P(x,y),\quad y'=Q(x,y),$$ 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 $\text{deg }P=\text{deg }Q=d$ having at least $$\tfrac{1}{2} \log_2 d-C\log_2\log_2 d $$ limit cycles. They also show that, for any nonnegative integer numbers $d,s_0,s_1,s_2,$ and $s_3$ satisfying the conditions $$ 2s_0+s_1+s_2+s_3=d^2,\quad |s_1+s_2-s_3|\leq d, $$ it is possible to construct a system with $\text{deg }P=\text{deg }Q=d$ having $2s_0$ imaginary singular points, $s_1$ attractors, $s_2$ repellers, and $s_3$ saddles.

MSC:
34C05Location of integral curves, singular points, limit cycles (ODE)
37C70Attractors and repellers, topological structure
34D45Attractors
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References:
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