A complete classification of quadratic differential systems according to the dimensions of $Aff(2,\mathbb R)$-orbits

In this article we consider the action of the group $Aff(2,\mathbb R)$ of affine transformations and time rescaling on real planar quadratic differential systems. Via affine invariant conditions we give a complete stratification of this family of systems according to the dimension $\mathcal D$ of affine orbits proving that $3\le\mathcal D\le6$. Moreover we give a complete topological classification of all the systems located on the orbits of dimension $\mathcal D\le5$ constructing the affine invariant criteria for the realization of each of 49 possible topologically distinct phase portraits.