  A theory for obtaining waveform for the effective entrainment of a weakly
forced oscillator is presented. Phase model analysis is combined with calculus
of variation to derive a waveform with which entrainment of an oscillator is
achieved with minimum power forcing signal. Optimal waveforms are calculated
from the phase response curve and a solution to a balancing condition. The
theory is tested in chemical entrainment experiments in which oscillations
close to and further away from a Hopf bifurcation exhibited sinusoidal and
higher harmonic nontrivial optimal waveforms, respectively.
