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We demonstrate that the modulation instability of the zone boundary mode in afinite (periodic) Fermi-Pasta-Ulam chain is the necessary but not sufficientcondition for the efficient energy transfer by localized excitations. Thistransfer results from the exclusion of complete energy exchange betweenspatially different parts of the chain, and the excitation level correspondingto that turns out to be twice more than threshold of zone boundary mode'sinstability. To obtain this result one needs in far going extension of thebeating concept to a wide class of finite oscillatory chains. In turn, such anextension leads to description of energy exchange and transition to energylocalization and transfer in terms of 'effective particles' and Limiting PhaseTrajectories. The 'effective particles' appear naturally when the frequencyspectrum crowding ensures the resonance interaction between zone boundary andtwo nearby nonlinear normal modes, but there are no additional resonances. Weshow that the Limiting Phase Trajectories corresponding to the most intensiveenergy exchange between 'effective particles' can be considered as analternative to Nonlinear Normal Modes, which describe the stationary process.

Limiting phase trajectories and the origin of energy localization in nonlinear oscillatory chains