  Weakly damped mechanical systems under small periodic forcing tend to exhibit
periodic response in a close vicinity of certain periodic orbits of their
conservative limit. Specifically, amplitude frequency plots for the
conservative limit have often been noted, both numerically and experimentally,
to serve as backbone curves for the near resonance peaks of the forced
response. In other cases, such a relationship between the unforced and forced
response was not observed. Here we provide a systematic mathematical analysis
that predicts which members of conservative periodic orbit families will serve
as backbone curves for the forced-damped response. We also obtain mathematical
conditions under which approximate numerical and experimental approaches, such
as energy balance and force appropriation, are justifiable. Finally, we derive
analytic criteria for the birth of isolated response branches (isolas) whose
identification is otherwise challenging from numerical continuation.
