A quadratic system with two parallel straight-line-isoclines

摘要

In this paper, a quadratic system with two parallel straight-line-isoclines is considered. Thissystem corresponds to the system of class II in the classification of Ye Yanqian [Ye Yanqianet al., Theory of Limit Cycles, Transl. Math. Monogr., vol. 66, American Mathematical Society,Providence, RI, 19861. Using the field rotation parameters of the constructed canonicalsystem and geometric properties of the spirals filling the interior and exterior domainsof its limit cycles, we prove that the maximum number of limit cycles in a quadraticsystem with two parallel straight-line-isoclines and two finite singular points is equal totwo. Besides, we obtain the same result in a different way: applying the Wintner-Perkotermination principle for multiple limit cycles and using the methods of global bifurcationtheory developed in [V.A. Gaiko, Global Bifurcation Theory and Hilbert's Sixteenth Problem,Kluwer, Boston, 20031.