In this paper we describe some recent advances in the basic theory and applications of phase space transport in nonlinear dynamic systems. These methods offer both qualitative and quantitative information about the behavior of solutions near homoclinic and heteroclinic motions in nonlinear dynamical systems. Applications of these ideas are found in fluid mixing and the escape of solutions from potential energy wells under the action of disturbances, for example, in models of ship capsize. In this work the theory is extended to a certain class of higher-order systems in which several time scales are involved. In addition, a new analytical estimate is derived and used for the rate of transport in the case of two-dimensional Poincare maps. Extensive simulation results from a specific ship dynamics model are used to demonstrate and verify these results.