# Entrainment of linear and non-linear systems

I'm trying to understand this paper: http://arxiv.org/abs/1301.2440 They use the Van der Pol oscillator model which gives oscillatory behaviour both sides of the bifurcation point: a limit cycle oscillator with an unstable fixed point and a noise-induced oscillator with a stable fixed point. The main claim is that when external periodic forcing is added to these systems, the noise-induced oscillator can only entrain to one-to-one ratio of internal-external frequency while the stable limit cycle oscillator can entrain to a number of different ratios.

What does entrainment actually mean? Does it mean that the internal frequency will become the same as the frequency of the external periodic function which is added to the internal system? If that's the case then for the one-to-one ratio case doesn't that mean that external and internal frequencies are the same? Then how is that entrainment if the frequencies were the same from before?

## 1 Answer

Briefly, suppose you have $$\dot x = f(x) + Ay(t)$$ where $x(t)$ has some intrinsic period $p_0$ (the internal frequency is $1/p_0$) when $A=0$, and $y(t) = y(t+q)$ has period $q$. Then you may get entrainment by increasing $A$.

Entrainment means that when the forcing term has period $q$, then $x(t)$ responds by oscillating at some ratio such as $p/q$ if the forcing strength $A$ is sufficiently large.

If the intrinsic frequency equals the forcing frequency you may not notice entrainment (although there could be phase synchronization), it is seen when the two frequencies differ.

• So essentially in a 1-1 entrainment the external and internal frequencies are approximately equal. As you increase the forcing strength the window of 'allowed'(for entrainment) external frequency gets bigger. Thanks for the answer. – argzar Mar 9 '13 at 22:38