Luo (Southern Illinois U. Edwardsville) presents a new theory for n-dimensional nonlinear dynamics. He presents the differential geometrical relations between two flows in two different dynamical systems through G-functions and then investigates the global transversality in nonlinear dynamical systems based on the G-functions. He discusses a theory for chaotic layer dynamics for nonlinear Hamiltonian systems, including the resonant theory of stochastic layers and the stochasticity of resonant layers. Nonlinear dynamics on the (2n-1)-dimensional equi-energy surface is also discussed, as is the stability and grazing bifurcations for dissipative nonlinear dynamical systems. Finally, switchability of a flow from a domain to an adjacent domain in discontinuous dynamical systems is discussed through the G-functions, and the first incremental increment for discontinuous dynamical systems are given.