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Mathematical Problems in Engineering
Volume 2010, Article ID 760479, 24 pages
Research Article

Limiting Phase Trajectories and Resonance Energy Transfer in a System of Two Coupled Oscillators

1N. N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, 4, Kosygina street, Moscow 119991, Russia
2Space Research Institute, Russian Academy of Sciences, Moscow 117997, Russia
3Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Champaign, IL 61820, USA

Received 30 July 2009; Accepted 6 November 2009

Academic Editor: José Balthazar

Copyright © 2010 L. I. Manevitch et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study a problem of energy exchange in a system of two coupled oscillators subject to 1 : 1 resonance. Our results exploit the concept of limiting phase trajectories (LPTs). The LPT, associated with full energy transfer, is, in certain sense, an alternative to nonlinear normal modes characterized by conservation of energy. We consider two benchmark examples. As a first example, we construct an LPT and examine the convergence to stationary oscillations for a Duffing oscillator subjected to resonance harmonic excitation. As a second example, we treat resonance oscillations in a system of two nonlinearly coupled oscillators. We demonstrate the reduction of the equations of motion to an equation of a single oscillator. It is shown that the most intense energy exchange and beating arise when motion of the equivalent oscillator is close to an LPT. Damped beating and the convergence to rest in a system with dissipation are demonstrated.