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