A detailed analysis of the periodic and chaotic aspects of the coupled set of nonlinear equations deduced by Stenflo is done. It is observed that period doubling takes place with increase of one of the parameters in the equation. The Hénon method is used to search for the periodic orbits, which is supported by an analysis of the power spectrum. The chaotic region is exhibited with the help of a Poincaré map, a power spectrum and its Lyapunov exponent. From the values of the Lyapunov exponents the fractal dimension of the chaotic system is estimated.