Dynamics are constructed for fractals utilizing the motion associated with Duffing equation. Using the paradigm of Fatou-Julia iteration, we develop iterations to map fractals accompanied with a criterion to ensure that the image is again a fractal. Because of the close link between mappings, differential equations and dynamical systems, one can introduce dynamics for fractals through differential equations such that they become points of the solution trajectory. There is no doubt that the differential equations have a distinct role for studying chaos. Therefore, characterization of fractals as trajectory points is an important step toward a better understanding of the link between chaos and fractal geometry. Moreover, it would be helpful to enhance and widen the scope of their applications in physics and engineering.