Iterated maps, which mimic the motion of nonlinear systems coupled to a bath, are described in the weak coupling regime. Classical time-dependent perturbation theory is used to derive the maps. To study vibrational energy transfer, the system is modeled by the harmonic and Morse oscillators. For chemical reactions the system is described by the double-well potential. Particular attention is given to the coupling between the systems and the bath, which is taken to be nonlinear in the bath modes. The maps provide a Very efficient way to numerically simulate the dynamics of the systems, but are unique in their ability to delineate the various coupling parameters that govern the dynamics. A simple "random phase" limit of the maps is discussed and leads to a kinetic description of the dynamics given by a multidimensional Fokker-Planck equation. Explicit expressions for the energy-diffusion coefficients are obtained.