In the paper under review, the authors study a dynamical systems approach to continued fractions. It is well-known that there is an interpretation of the continued fraction expansion process as a discrete two-dimensional dynamical system. On the contrary, in the present paper the authors study a new three-dimensional dynamical system which can be used to model the continued fraction expansion of quadratic irrationals, by this means continuing their research from [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 12, e-Suppl., e2136--e2151 (2009; Zbl 1239.11009)]. Simply speaking, the three-dimensional points correspond to the coefficients of quadratic polynomials, which result from the continued fraction algorithm. After studying the properties of such dynamical systems, the paper is concluded with a section discussing the connection of these results with the solutions of certain Pell equations.