Why is $\cos((\omega+\alpha\cos(\omega' t))t)$ the wrong model for frequency modulation? So I was trying to program vibrato, or freqency modulation, naively using the model:
$$\cos((\omega + \alpha\cos(\omega' t))t)$$
Where $\alpha \lt \omega$  and $\omega' \ll \omega$. For practical purposes, assume $\omega=200\,\mathrm{Hz},\ \alpha=5\,\mathrm{Hz},\ \omega'=1\,\mathrm{Hz}$ so that
$$195\,\mathrm{Hz}<\omega + \alpha\cos(\omega' t)<205\,\mathrm{Hz}.$$
However in practice this looks and sounds completely wrong. It starts out sounding like vibrato, but the modulation amplitude increases indefinitely:

After some Googling, I find that I'm supposed to use the model:
$$\cos(\theta(t))$$
where in general,
$$\omega(t)=\frac{d\theta(t)}{dt}$$
So I set it up as:
$$\theta(t)=\int\omega + \alpha\cos(\omega' t)\,dt$$
and I get the desired result:

Question is, being that I'm a little rusty with math, how can the first approach be analyzed to show why it behaves the way it does?

Let $\theta(t) = \omega t + t \alpha \cos (\xi t)$, then
$\theta'(t) = \omega + \alpha \cos (\xi t) - t \alpha \xi \sin (\xi t)$, so you
can see that the instantaneous frequency is unbounded.
