We propose an adequate analytical approach to the description of nonlinear vibration with strong energy exchange between weakly coupled oscillators. The fundamental notion of the limiting phase trajectory (LPT) corresponding to full energy transfer is introduced. In a certain sense this is alternative to the nonlinear normal mode (NNM) characterized by conservation of energy. Many well-known approximations based on NNMs turn out to be valid for the case of weak energy exchange, and the proposed approach can be used for the description of nonlinear processes with strong energy exchange between weakly coupled oscillators or oscillatory chains. Such a description is formally similar to that of a vibro-impact process and can be considered as generating such a process when dealing with a description of other processes with intensive energy transfer. We consider subsequently two weakly coupled conservative, damped and forced nonlinear oscillators as well as a system of two weakly coupled oscillatory chains. It is shown that two dynamic transitions occur in the system under consideration when the nonlinearity parameter increases. The first, well-known, transition corresponds to the bifurcation of one of the cooperative nonlinear normal modes (NNMs) and leads to its instability and the appearance of two stable normal modes with predominant excitation of one of oscillators. The second transition is characterized by the instability of the the limiting phase trajectory (LPT), corresponding to complete energy transfer between the oscillators. As a result of such a transition the complete energy transfer is broken, and energy localization on one of oscillators (whereas it was initially concentrated on it) becomes possible instead of beating. This motion with the most intensive energy transfer turns out to be as important as NNMs for the explanation of dynamic transformations and the description of closely related regimes.