Electronic Journal of Diﬀerential Equations, Vol. 2017 (2017), No. 295, pp. 1–122. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GEOMETRIC AND ALGEBRAIC CLASSIFICATION OF QUADRATIC DIFFERENTIAL SYSTEMS WITH INVARIANT HYPERBOLAS REGILENE D. S. OLIVEIRA, ALEX C. REZENDE, DANA SCHLOMIUK, NICOLAE VULPE Abstract. Let QSH be the whole class of non-degenerate planar quadratic diﬀerential systems possessing at least one invariant hyperbola. We classify this family of systems, modulo the action of the group of real aﬃne transfor- mations and time rescaling, according to their geometric properties encoded inthe conﬁgurations of invariant hyperbolas and invariant straight lines which these systems possess. The classiﬁcation is given both in terms of algebraic geometric invariants and also in terms of aﬃne invariant polynomials. It yields a total of 205 distinct such conﬁgurations. We have 162 conﬁgurations for the subclass QSH (η>0) of systems which possess three distinct real singularities at inﬁnity in P 2 (C), and 43 conﬁgurations for the subclass QSH (η=0) of systems which possess either exactly two distinct real singularities at inﬁnity or the line at inﬁnity ﬁlled up with singularities. The algebraic classiﬁcation, based on the invariant polynomials, is also an algorithm which makes it possible to verify for any given real quadratic diﬀerential system if it has invariant hy- perbolas or not and to specify its conﬁguration of invariant hyperbolas and straight lines. Contents 1. Introduction and statement of the main results 2 2. Basic concepts and auxiliary results 8 3. Conﬁgurations of invariant hyperbolas for the class QSH (η>0) 25 3.1. Subcase θ =0 30 3.2. Subcase θ =0 76 4. Conﬁgurations of invariant hyperbolas for the class QSH (η=0) 97 4.1. Possibility M (˜ a,x,y) =0 98 4.2. Possibility M (˜ a,x,y)=0= C 2 (˜ a,x,y) 115 5. Concluding comments 117 5.1. Concluding comments for η> 0 117 5.2. Concluding comments for η =0 118 Acknowledgments 120 References 120 2010 Mathematics Subject Classiﬁcation. 34C23, 34A34. Key words and phrases. Quadratic diﬀerential systems; algebraic solution; conﬁguration of algebraic solutions; aﬃne invariant polynomials; group action. c 2017 Texas State University. Submitted February 9, 2017. Published November 28, 2017. 1