  Theoretical models that describe oscillations in biological systems are often
either a limit cycle oscillator, where the deterministic nonlinear dynamics
gives sustained periodic oscillations, or a noise-induced oscillator, where a
fixed point is linearly stable with complex eigenvalues and addition of noise
gives oscillations around the fixed point with fluctuating amplitude. We
investigate how each class of model behaves under the external periodic
forcing, taking the well-studied van der Pol equation as an example. We find
that, when the forcing is additive, the noise-induced oscillator can show only
one-to-one entrainment to the external frequency, in contrast to the limit
cycle oscillator which is known to entrain to any ratio. When the external
forcing is multiplicative, on the other hand, the noise-induced oscillator can
show entrainment to a few ratios other than one-to-one, while the limit cycle
oscillator shows entrain to any ratio. The noise blurs the entrainment in
general, but clear entrainment regions for limit cycles can be identified as
long as the noise is not too strong.
