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Cyclicity of graphics with semi-hyperbolic points inside quadratic systems. (English) Zbl 0989.37047

Summary: This paper is a part of the proof of the existential part of Hilbert’s 16th problem for quadratic vector fields. Its principal aim is the proof of the finite cyclicity of four elementary graphics among quadratic systems with two semi-hyperbolic points and surrounding a center, namely, the graphics \((I_5^2)\), \((I_{16b}^2)\), \((H_9^1)\), and \((H_{11}^1)\). The technique used is a refinement of the technique of Khovanskii as adapted to the finite cyclicity of graphics by Il’yashenko and Yakovenko, together with an equivalent of the Bautin trick to treat the center case. We show that the cyclicity of each of the first three graphics is equal to 2 and that the cyclicity of the fourth one is equal to three. We improve the known results about finite cyclicity of the graphics \((H_8^1)\), \((H_{10}^1)\), \((I_4^2)\) by showing that their cyclicities are equal to 2.

MSC:

37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37G20 Hyperbolic singular points with homoclinic trajectories in dynamical systems
34C23 Bifurcation theory for ordinary differential equations
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