Currently, the dynamic response of nonlinear structures is generally determined by numerical integration of the incremental coupled second order differential equations of motion involving their tangent stiffness matrices. However, the dominant paradigm being computational, the initial value problems (IVPs) being solved are rarely stated in an analytical form. Three new formulations of analytical IVPs for nonlinear structures are presented in this Paper. Firstly, it is argued that only the IVPs based upon coupled third order differential equations of motion are proper for nonlinear MDOF dynamical systems. Secondly, it is shown that, for homogeneous dynamical systems, coupled second order differential equations are applicable. Finally, for general nonlinear dynamical systems, it is proposed to formulate the IVPs using decoupled nonlinear second order differential equations. Theoretical significance of these three analytical prospects in nonlinear structural dynamics is discussed.