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 a 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 there corresponds a finite open segment \(I\subset \mathbb {R}^4\) . A quadrangle in F that corresponds to the midpoint of I is called a balanced quadrangle. Let M be the set of balanced quadrangles. The function \(f:M\rightarrow M\) is defined in the following way: angles of the balanced quadrangle \(Q'\) , \(Q'=f(Q)\) , 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.
