In this work we show how the composition of maps allows us to multiply, enlarge and move stable domains in phase and parameter spaces of discrete nonlinear systems. Using Hénon maps with distinct parameters we generate many identical copies of isoperiodic stable structures (ISSs) in the parameter space and attractors in phase space. The equivalence of the identical ISSs is checked by the largest Lyapunov exponent analysis and the multiplied basins of attraction become riddled. Our proliferation procedure should be applicable to any two-dimensional nonlinear system. In high-dimensional dynamical systems the possibility of controlling the dynamics through parametric changes is of great interest. Adding a time dependent parameter it is possible to control the dynamics of the paradigmatic Hénon map generating multiply Isoperiodic Stable Structures (ISSs) on the parameter space and consequently increasing the number of attractors on the phase space. Numerical simulations and analytical results explain the origin of new stable domains due to saddle-node bifurcations for specific parametric combinations. The distance between the multiplied ISSs can be controlled by the intensity of the time dependent parameter and general rules for the occurrence of proliferation are treated in details. We believe that the present study represents a significantly new insight on the use of alternating forces to control the dynamics of complex nonlinear systems modeled by twodimensional maps and can be extended to various applications ranging from physics, biology to engineering.