  We present an adequate analytical approach to the description of nonlinear
vibration with strong energy exchange between weakly coupled oscillators and
oscillatory chains. The fundamental notion of the limiting phase trajectory
(LPT) corresponding to complete energy exchange is introduced. At first we
propose a simple analytical description of vibrations of nonlinear oscillators.
We show that two dynamical transitions occur in the system. First of them
corresponds to the bifurcation of anti-phase vibrations of oscillators. And the
second one is caused by coincidence of LPT with separatrix dividing two stable
stationary states and leads to qualitative change in both phase and temporal
behavior of the LPT. Next problem under consideration relates to intensive
intermodal exchange in the periodic nonlinear systems with finite (n>2) number
of degrees of freedom. We consider two limiting cases. If the number of
particles is not large enough, the energy exchange between nonlinear normal
modes in two-dimensional integral manifolds is considered. When the number of
the particles increases the energy exchange between neighbor integral manifolds
becomes important that leads to formation of the localized excitations
resembling the breathers in the one-dimensional continuum media.
