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We present an adequate analytical approach to the description of nonlinearvibration with strong energy exchange between weakly coupled oscillators andoscillatory chains. The fundamental notion of the limiting phase trajectory(LPT) corresponding to complete energy exchange is introduced. At first wepropose a simple analytical description of vibrations of nonlinear oscillators.We show that two dynamical transitions occur in the system. First of themcorresponds to the bifurcation of anti-phase vibrations of oscillators. And thesecond one is caused by coincidence of LPT with separatrix dividing two stablestationary states and leads to qualitative change in both phase and temporalbehavior of the LPT. Next problem under consideration relates to intensiveintermodal exchange in the periodic nonlinear systems with finite (n>2) numberof degrees of freedom. We consider two limiting cases. If the number ofparticles is not large enough, the energy exchange between nonlinear normalmodes in two-dimensional integral manifolds is considered. When the number ofthe particles increases the energy exchange between neighbor integral manifoldsbecomes important that leads to formation of the localized excitationsresembling the breathers in the one-dimensional continuum media.

Resonant energy exchange in nonlinear oscillatory chains and Limiting Phase Trajectories: from small to large systems