Dynamics of a family of maps defined by quadratic polynomials

We consider maps $F \colon {\mathbb R}^{2} \rightarrow \mathbb R^{2}$, whose coordinates are homogeneous polynomials in $\mathbb R[x, y]$ of degree $2$. These maps send lines passing through the origin into lines passing through the origin. Our goal is to study how these lines are moved under the action of $F$. We show that there is a real analytic variety $\mathcal{F}^{2}$, where two sets can be clearly distinguished. One set $\mathcal{U} \subseteq \mathcal{F}^{2}$ is made up of transformations that have "hidden hyperbolic" dynamics, and its complement $\mathcal{F}^{2} \setminus \mathcal{U}$ contains maps that show a chaotic behavior.