Limiting Phase Trajectories as an Alternative to Nonlinear Normal Modes ☆

Research paper by I. Manevitch Leonid, P. Koroleva (Kikot) Irina

Indexed on: 29 Mar '16Published on: 28 Mar '16Published in: Procedia IUTAM


We discuss a recently developed concept of limiting phase trajectories (LPTs) allowing a unified description of resonance, highly non-stationary processes for a wide range of classical and quantum dynamical systems with constant and varying parameters. This concept provides a far going extension and adequate mathematical description of the well-known linear beating phenomenon to a diverse variety of nonlinear systems ranging from classical multi-particle models to nonlinear quantum tunneling. While stationary (and non-stationary, but non-resonant) oscllations can be described in the framework of non-linear normal modes (NNMs) concept, it is not so in the considered case of resonant non-stationary processes. In the latter case which is characterized by intense energy exchange between different parts of a system, an additiional slow time scale appears. The energy exchange proceeds in this time scale and can be identified as strong modulation of the fast oscillations. The aforementioned resonant non-staionary prcesses include, e.g., targeted energy transfer, non-stationary vibrations of carbon nanotubes, quantum tunneling, auto-resonance and non-conventional synchronization. Besides the non-linear beating, the LPT concept allows one to find the conditions of transition from intense energy exchange to strongly localized (e.g. breather-like) excitations. A special mathematical technique based on the non- smooth temporal transformations leads to the clear and simple description of strongly modulated regimes. The role of LPTs in the theory of resonance non-stationary processes turns out to be similar to that of NNMs in stationary case.