Integrability and geometry of quadratic differential systems with invariant hyperbolas - Travaglini, Ana Maria (2021)
Planar polynomial differential systems occur very often in various branches of applied mathematics, in modeling natural phenomena, in astrophysics, in the equations of continuity describing the interactions of ions, electrons and neutral species in plasma physics, among other situations. Such differential systems have also theoretical importance. Several problems stated more than one hundred years ago on polynomial differential systems are still open, for instance, the second part of Hilberts 16th problem stated by Hilbert in (HILBERT, 1902), the problem of algebraic integrability stated by Poincaré in (POINCARÉ, 1891a), (POINCARÉ, 1891b), problems on integrability resulting from the work of Darboux (DARBOUX, 1878) and the problem of the center also stated by Poincaré (POINCARÉ, 1885). They are still unsolved, except for the problem of the center solved only in the quadratic case. In this thesis we denote by QSH be the whole class of non-degenerate planar quadratic differential systems possessing at least one invariant hyperbola. QSH is a rich family of systems displaying various kinds of integrability: polynomial, algebraic (rational), Darboux, generalized Darboux, Liouvillian. The goal of this investigation is to study this class from the viewpoint of the theory of Darboux: To separate the integrable system in QSH, to classify them according to the kind of first integral they possess and study their geometry. Our main motivation and goal, apart from gathering data, is to study the relationship between integrability and the geometry of the systems as expressed in their configurations of invariant algebraic curves, to study the bifurcations of their configurations as well as their relations with the bifurcations of the phase portraits.