Affine invariant criteria for a quadratic system to possess an invariant conic and a Darboux invariant [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Nicolae Vulpe
Institute of Mathematics and Computer Science


It is known that on the set $\mathbf{QS}$ of all quadratic differential systems acts the group $Aff(2,{\mathbb R})$ of real affine transformations on the plane. Using this group action in the paper "Quadratic systems with an invariant conic having Darboux invariants" (by J. Llibre and R. Oliveira, preprint 2015), the authors completed the characterization of the phase portraits in the Poincaré disc of all planar quadratic polynomial differential systems having an invariant conic $\mathcal{C}: f(x; y) = 0$, and a Darboux invariant of the form $f(x; y)e^{st}$ with $s\in \R\setminus\{0\}$ (here $t$ is the time). We denote this class of quadratic systems by $\mathbf{QS}^{\mathcal{C}}_{\mathcal{DI}}$. Applying the algebraic theory of invariants of differential equations (elaborated by C. Sibirsky) we present a complete classification of of the class $\mathbf{QS}^{\mathcal{C}}_{\mathcal{DI}}$. First we detect necessary and sufficient conditions for an arbitrary quadratic system to be in this class and, secondly, we construct affine invariant criteria for the realization of each one of the possible phase portraits of the systems in $\mathbf{QS}^{\mathcal{C}}_{\mathcal{DI}}$.