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The main objective of this paper is to find necessary and sufficient conditions for a 1:−4 resonant system of the formx˙=x−a10x2−a01xy−a12y2,y˙=−4y+b2,−1x2+b10xy+b01y2 to have a center at the origin. Since applying a linear change of variables any system of this form can be transformed either to system with a10=1 or a10=0, these are the two cases considered here. When a10=1 there appear 46 resonant...
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