Analytic Conditions for Targeted Energy Transfer between Nonlinear Oscillators or Discrete Breathers

It is well known that any amount of energy injected in a harmonic oscillator which is resonant and weakly coupled with a second harmonic oscillator, tunnels back and forth between these two oscillators. When the two oscillators are anharmonic, the amplitude dependence of their frequencies breaks, in general, any eventual initial resonance so that no substantial energy transfer occurs unless, exceptionally, an almost perfect resonance persists. This paper considers this interesting situation more generally between two discrete breathers belonging to two weakly coupled nonlinear systems, finite or infinite. A specific amount of energy injected as a discrete breather in a nonlinear system (donor) which is weakly coupled to another nonlinear system (acceptor) sustaining another discrete breather, might be totally transferred and oscillate back and forth between these donor and acceptor breathers. The condition is that a certain well defined detuning function is bounded from above and below by two coupling functions. This targeted energy transfer is selective, i.e., it only occurs for an initial energy close to a specific value. The explicit calculation of these functions in complex models with numerical techniques developed earlier for discrete breathers, allows one to detect the existence of possible targeted energy transfer, between which breathers, and at which energy. It should also help for designing models having desired targeted energy transfer properties at will. We also show how extra linear resonances could make the energy transfer incomplete and irreversible. Future developments of the theory will be able to describe more spectacular effects, such as targeted energy transfer cascades and avalanches, and energy funnels. Besides rather short term applications for artificially built devices, this theory might provide an essential clue for understanding puzzling problems of energy kinetics in real materials, chemistry, and bioenergetics. Preprint submitted to Physica B August 17 2000