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

Entrainment of noise-induced and limit cycle oscillators under weak noise