A dynamical system in the space of convex quadrangles

Let us consider a family $F(\alpha ,\beta ,\gamma ,\delta )$ of convex quadrangles in the plane with given angles $\{\alpha ,\beta ,\gamma ,\delta \}$ and with the perimeter $2\pi$. Such quadrangle $Q\in F(\alpha ,\beta ,\gamma ,\delta )$ can be considered as a point $(x_1,x_2,x_3,x_4)\in {\mathbb{R}}^4$, where $\{x_1,x_2,x_3,x_4\}$ are lengths of edges. Then to $F$ a finite open segment $I\subset {\mathbb{R}}^4$ is corresponded. A quadrangle in $F$, that corresponds to the midpoint of $I$ is called a \emph{balanced quadrangle}. Let $M$ be the set of balanced quadrangles. The function $f:M\to M$ is defined in the following way: angles of the balanced quadrangle $Q^{\mathrm{\prime }}$, $Q^{\mathrm{\prime }}=f(Q)$, are numerically equal to edges of $Q$. The map $f$ defines a dynamical system in the space of balanced quadrangles. In this work we study properties of this system.