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This Report discusses a recently developed concept of Limiting PhaseTrajectories (LPTs) providing a unified description of resonant energytransport in a wide range of classical and quantum dynamical systems withconstant and time-varying parameters. It is shown that strongly modulatednon-stationary processes occurring in a nonlinear oscillator array undercertain initial conditions may characterize either maximum possible energyexchange between the clusters of oscillators (effective particles) or maximumenergy transfer from an external source of energy to the system. Thetrajectories corresponding to these processes are referred to as LPTs. Thedevelopment and the use of the LPT concept are motivated by the fact thathighly non-stationary resonance processes occurring in a broad variety offinite-dimensional physical models are beyond the well-known paradigm of thenonlinear normal modes (NNMs), fully justified only for stationary processes,their stability and bifurcations, as well as for non-stationary processesdescribed approximately by some combinations of non-resonating normal modes.Thus, the role of LPTs in understanding and description of non-stationaryenergy transfer is similar to the role of NNMs for the stationary processes.Several applications of the LPT concept to significant nonlinear problems and ascenario of the transition from regular to chaotic behavior with the LPTsimplication are presented. In order to highlight the novelty and perspectivesof the developed approach, we place the LPT concept into the context of complexdynamical phenomena related to energy transfer problems.

Limiting Phase Trajectories: a new paradigm for the study of highly non-stationary processes in Nonlinear Physics